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/ 4 < (g.edgeset.ncard : r) \n "},{"id":"erdos_1034_erdos_1034_1","label":"Erdős 1034: erdos_1034","status":"proved","category":"research","downstream_reach":4,"bridge_distance":0.0,"coverage_per_cost":2.4,"q":"erdos 1034: erdos_1034 erdosproblems/1034.lean r) (h : 0 < ) :\n f (n : n) in attop, g : simplegraph (fin n),\n (n : r) ^ 2 / 4 < (g.edgeset.ncard : r) \n "},{"id":"erdos_1034_erdos_1034_2","label":"Erdős 1034: erdos_1034","status":"proved","category":"research","downstream_reach":4,"bridge_distance":0.0,"coverage_per_cost":2.4,"q":"erdos 1034: erdos_1034 erdosproblems/1034.lean r) (h : 0 < ) :\n f (n : n) in attop, g : simplegraph (fin n),\n (n : r) ^ 2 / 4 < (g.edgeset.ncard : r) \n "},{"id":"erdos_1034_erdos_1034_3","label":"Erdős 1034: erdos_1034","status":"proved","category":"research","downstream_reach":4,"bridge_distance":0.0,"coverage_per_cost":2.4,"q":"erdos 1034: erdos_1034 erdosproblems/1034.lean r) (h : 0 < ) :\n f (n : n) in attop, g : simplegraph (fin n), 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(g.neighborset v).ncard = d}.ncard "},{"id":"erdos_1038_erdos_1038_0","label":"Erdős 1038: erdos_1038","status":"open","category":"research","downstream_reach":9,"bridge_distance":1.0,"coverage_per_cost":1.35,"q":"erdos 1038: erdos_1038 erdosproblems/1038.lean erdos_1038.parts.i.answer =\n f : {f : polynomial r // f.monic f = 1 \n (f.roots.filter fun x => x set.icc (-"},{"id":"erdos_1038_erdos_1038_1","label":"Erdős 1038: erdos_1038","status":"proved","category":"research","downstream_reach":9,"bridge_distance":0.0,"coverage_per_cost":5.4,"q":"erdos 1038: erdos_1038 erdosproblems/1038.lean 2 * 2 ^ (1 / 2 : r) =\n f : {f : polynomial r // f.monic \n (f.roots.filter fun x => x set.icc (-1 : r) 1).card"},{"id":"erdos_1038_erdos_1038_2","label":"Erdős 1038: erdos_1038","status":"proved","category":"research","downstream_reach":9,"bridge_distance":0.0,"coverage_per_cost":5.4,"q":"erdos 1038: erdos_1038 erdosproblems/1038.lean f : {f : polynomial r // f.monic f = 1 \n (f.roots.filter fun x => x set.icc (-1 : r) 1).card = f.natdegree},\n "},{"id":"erdos_1038_erdos_1038_3","label":"Erdős 1038: erdos_1038","status":"proved","category":"research","downstream_reach":9,"bridge_distance":0.0,"coverage_per_cost":5.4,"q":"erdos 1038: erdos_1038 erdosproblems/1038.lean 2 ^ (4 / 3 : r) - 1 \n f : {f : polynomial r // f.monic f = 1 \n (f.roots.filter fun x => x set.icc (-1 : r)"},{"id":"erdos_1038_erdos_1038_4","label":"Erdős 1038: erdos_1038","status":"proved","category":"textbook","downstream_reach":9,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 1038: erdos_1038 erdosproblems/1038.lean r) (hd : |d| 1) :\n volume {x : r | |(x - m) ^ 2 - d ^ 2| < 1} =\n ennreal.ofreal (2 * real.sqrt (1 + d ^ 2))"},{"id":"erdos_1038_erdos_1038_5","label":"Erdős 1038: erdos_1038","status":"proved","category":"textbook","downstream_reach":9,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 1038: erdos_1038 erdosproblems/1038.lean r) (ha : |a| 1) (hb : |b| 1) :\n volume {x : r | 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= 0"},{"id":"erdos_158_erdos_158_1","label":"Erdős 158: erdos_158","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 158: erdos_158 erdosproblems/158.lean set n} (hainf : a.infinite) (hasid : issidon a) :\n liminf (fun n ennreal.ofreal ((a .iio n).ncard * n ^ (- 1 / 2 "},{"id":"erdos_158_erdos_158_2","label":"Erdős 158: erdos_158","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 158: erdos_158 erdosproblems/158.lean set n} (hainf : a.infinite) (hasid : issidon a) :\n liminf (fun n : n => (a .iio n).ncard * (n : r) ^ (- 1 / 2 : r))"},{"id":"erdos_159_erdos_159_0","label":"Erdős 159: erdos_159","status":"open","category":"research","downstream_reach":1,"bridge_distance":2.0,"coverage_per_cost":0.1,"q":"erdos 159: erdos_159 erdosproblems/159.lean (c : r) (_ : 0 < c) (c : r),\n (n : n), 1 n \n (simplegraph.graphramsey (simplegraph.cyclegraph 4)\n 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n) => exp (c * log (n : r) ^ ((1 : r) / 12)))\n =o[attop] fun n => (erdos_160.h n : r)"},{"id":"erdos_161_erdos_161_0","label":"Erdős 161: erdos_161","status":"open","category":"research","downstream_reach":2,"bridge_distance":2.0,"coverage_per_cost":0.2,"q":"erdos 161: erdos_161 erdosproblems/161.lean t : n, 1 t \n {p | p.1 set.ico (0 : r) (1 / 2) p.2 set.ico (0 : r) (1 / 2) \n (fun n (hypergraph"},{"id":"erdos_161_erdos_161_1","label":"Erdős 161: erdos_161","status":"open","category":"research","downstream_reach":2,"bridge_distance":2.0,"coverage_per_cost":0.2,"q":"erdos 161: erdos_161 erdosproblems/161.lean t : n, 3 t : r,\n 0 < < 1 / 2 0 < < 1 / 2 \n (fun n (hypergraph.balancedcoloringthresho"},{"id":"erdos_162_erdos_162_0","label":"Erdős 162: erdos_162","status":"open","category":"research","downstream_reach":5,"bridge_distance":1.0,"coverage_per_cost":0.75,"q":"erdos 162: erdos_162 erdosproblems/162.lean r) (h0 : 0 ) (h1 : < 1 / 2) :\n c : r, (fun n : n (f n : r)) ~[attop] (fun n 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erdos_171","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 171: erdos_171 erdosproblems/171.lean : r, 0 < t : n, 1 t f n : n in attop,\n a : finset (fin n fin t), * t ^ n a.card \n l : "},{"id":"erdos_172_erdos_172_0","label":"Erdős 172: erdos_172","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 172: erdos_172 erdosproblems/172.lean (n : n) (color : n fin n) (m), (a : finset n), a.card m c, (s : finset a),\n s.nonempty color ( x s"},{"id":"erdos_173_erdos_173_0","label":"Erdős 173: erdos_173","status":"open","category":"research","downstream_reach":3,"bridge_distance":1.0,"coverage_per_cost":0.45,"q":"erdos 173: erdos_173 erdosproblems/173.lean (c : r2 fin 2) (t1 t2 : triangle r r2),\n hasmonochromaticcopy c t1 hasmonochromaticcopy c t2 \n : "},{"id":"erdos_173_erdos_173_1","label":"Erdős 173: erdos_173","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 173: erdos_173 erdosproblems/173.lean r2 fin 2) (t : triangle r r2)\n (ht : (t.points 0) (t.points 1) (t.points 2) = real.pi / 2) :\n hasmonochromatic"},{"id":"erdos_173_erdos_173_2","label":"Erdős 173: erdos_173","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 173: erdos_173 erdosproblems/173.lean (c : r2 fin 2) (t : triangle r r2),\n dist (t.points 0) (t.points 1) = dist (t.points 1) (t.points 2) \n dist ("},{"id":"erdos_174_isRamsey_singleton_0","label":"Erdős 174: isRamsey_singleton","status":"open","category":"test","downstream_reach":8,"bridge_distance":1.0,"coverage_per_cost":1.2,"q":"erdos 174: isramsey_singleton erdosproblems/174.lean n} (x : euclideanspace r (fin n)) : isramsey {x}"},{"id":"erdos_174_erdos_174_1","label":"Erdős 174: erdos_174","status":"open","category":"research","downstream_reach":8,"bridge_distance":1.0,"coverage_per_cost":1.2,"q":"erdos 174: erdos_174 erdosproblems/174.lean n) :\n {a : set (euclideanspace r (fin n)) | isramsey a} = erdos_174.answer n"},{"id":"erdos_174_erdos_174_2","label":"Erdős 174: erdos_174","status":"proved","category":"research","downstream_reach":8,"bridge_distance":0.0,"coverage_per_cost":4.8,"q":"erdos 174: erdos_174 erdosproblems/174.lean n} (a : set (euclideanspace r (fin n)))\n (ha : isramsey a) : euclideangeometry.cospherical a"},{"id":"erdos_174_erdos_174_3","label":"Erdős 174: erdos_174","status":"open","category":"research","downstream_reach":8,"bridge_distance":1.0,"coverage_per_cost":1.2,"q":"erdos 174: erdos_174 erdosproblems/174.lean n} (a : set (euclideanspace r (fin n)))\n (ha : a.finite) (hs : euclideangeometry.cospherical a) : isramsey a"},{"id":"erdos_174_erdos_174_4","label":"Erdős 174: erdos_174","status":"open","category":"research","downstream_reach":8,"bridge_distance":1.0,"coverage_per_cost":1.2,"q":"erdos 174: erdos_174 erdosproblems/174.lean n}\n (a : set (euclideanspace r (fin n))) (ha : a.finite) :\n isramsey a issubtransitive a"},{"id":"erdos_174_erdos_174_5","label":"Erdős 174: erdos_174","status":"proved","category":"research","downstream_reach":8,"bridge_distance":0.0,"coverage_per_cost":4.8,"q":"erdos 174: erdos_174 erdosproblems/174.lean n} (a : fin n r) :\n isramsey {x : euclideanspace r (fin n) | i, x i = 0 x i = a i}"},{"id":"erdos_174_erdos_174_6","label":"Erdős 174: erdos_174","status":"proved","category":"research","downstream_reach":8,"bridge_distance":0.0,"coverage_per_cost":4.8,"q":"erdos 174: erdos_174 erdosproblems/174.lean n} (a : set (euclideanspace r (fin n)))\n (ha : affineindependent r (subtype.val : a euclideanspace r (fin n))) :\n "},{"id":"erdos_174_erdos_174_7","label":"Erdős 174: erdos_174","status":"proved","category":"research","downstream_reach":8,"bridge_distance":0.0,"coverage_per_cost":4.8,"q":"erdos 174: erdos_174 erdosproblems/174.lean r) (hh : h = 0) :\n isramsey ({!2[-s, 0], !2[s, 0], !2[-t, h], !2[t, h]} :\n set (euclideanspace r (fin 2)))"},{"id":"erdos_175_erdos_175_0","label":"Erdős 175: erdos_175","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 175: erdos_175 erdosproblems/175.lean n) (hn : 5 n) :\n squarefree ((2 * n).choose n)"},{"id":"erdos_178_erdos_178_0","label":"Erdős 178: erdos_178","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 178: erdos_178 erdosproblems/178.lean (a : n n n) (ha : i, strictmono (a i)),\n f : n z, ( n, f n = 1 f n = -1) \n d : n, c : n, m"},{"id":"erdos_18_practicalH_one_0","label":"Erdős 18: practicalH_one","status":"proved","category":"test","downstream_reach":13,"bridge_distance":0.0,"coverage_per_cost":7.8,"q":"erdos 18: practicalh_one erdosproblems/18.lean practicalh 1 = 1"},{"id":"erdos_18_practicalH_two_1","label":"Erdős 18: practicalH_two","status":"proved","category":"test","downstream_reach":13,"bridge_distance":0.0,"coverage_per_cost":7.8,"q":"erdos 18: practicalh_two erdosproblems/18.lean practicalh 2 = 1"},{"id":"erdos_18_practicalH_six_2","label":"Erdős 18: practicalH_six","status":"proved","category":"test","downstream_reach":13,"bridge_distance":0.0,"coverage_per_cost":7.8,"q":"erdos 18: practicalh_six erdosproblems/18.lean practicalh 6 = 2"},{"id":"erdos_18_practicalH_twelve_3","label":"Erdős 18: practicalH_twelve","status":"proved","category":"test","downstream_reach":13,"bridge_distance":0.0,"coverage_per_cost":7.8,"q":"erdos 18: practicalh_twelve erdosproblems/18.lean practicalh 12 = 3"},{"id":"erdos_18_practicalH_fiftysix_4","label":"Erdős 18: practicalH_fiftysix","status":"open","category":"test","downstream_reach":13,"bridge_distance":1.0,"coverage_per_cost":1.95,"q":"erdos 18: practicalh_fiftysix erdosproblems/18.lean practicalh 56 = 5"},{"id":"erdos_18_practicalH_7200_5","label":"Erdős 18: practicalH_7200","status":"open","category":"test","downstream_reach":13,"bridge_distance":1.0,"coverage_per_cost":1.95,"q":"erdos 18: practicalh_7200 erdosproblems/18.lean practicalh 7200 = 6"},{"id":"erdos_18_practicalH_le_divisors_6","label":"Erdős 18: practicalH_le_divisors","status":"proved","category":"test","downstream_reach":13,"bridge_distance":0.0,"coverage_per_cost":7.8,"q":"erdos 18: practicalh_le_divisors erdosproblems/18.lean n) (hn : nat.ispractical n) :\n practicalh n n.divisors.card"},{"id":"erdos_18_factorial_isPractical_7","label":"Erdős 18: factorial_isPractical","status":"proved","category":"textbook","downstream_reach":13,"bridge_distance":0.0,"coverage_per_cost":10.4,"q":"erdos 18: factorial_ispractical 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r)))"},{"id":"erdos_323_erdos_323_3","label":"Erdős 323: erdos_323","status":"open","category":"research","downstream_reach":4,"bridge_distance":1.0,"coverage_per_cost":0.6,"q":"erdos 323: erdos_323 erdosproblems/323.lean k > 2, (fun (x : n) (f k k x : r)) =o[attop] (fun (x : n) (x : r))"},{"id":"erdos_324_erdos_324_0","label":"Erdős 324: erdos_324","status":"open","category":"research","downstream_reach":2,"bridge_distance":2.0,"coverage_per_cost":0.2,"q":"erdos 324: erdos_324 erdosproblems/324.lean f : z[x], {(a, b) : n n | a < b}.injon fun (a, b) => f.eval (a : z) + f.eval (b : z)"},{"id":"erdos_324_erdos_324_1","label":"Erdős 324: erdos_324","status":"open","category":"research","downstream_reach":2,"bridge_distance":2.0,"coverage_per_cost":0.2,"q":"erdos 324: erdos_324 erdosproblems/324.lean {(a, b) : n n | a < b}.injon fun (a, b) => a ^ 5 + b ^ 5"},{"id":"erdos_325_erdos_325_0","label":"Erdős 325: erdos_325","status":"open","category":"research","downstream_reach":3,"bridge_distance":1.0,"coverage_per_cost":0.45,"q":"erdos 325: erdos_325 erdosproblems/325.lean k : n, 3 k (fun x : n => (x : r) ^ (3 / k : r)) =o[attop]\n (fun x : n => (cardissumthreepowerbelow k x : r))"},{"id":"erdos_325_erdos_325_1","label":"Erdős 325: erdos_325","status":"open","category":"research","downstream_reach":3,"bridge_distance":1.0,"coverage_per_cost":0.45,"q":"erdos 325: erdos_325 erdosproblems/325.lean > 0, k : n, 3 k (fun x : n => (x : r) ^ ((3 / k : r) - )) =o[attop]\n (fun x => (cardissumthreepowerbelow"},{"id":"erdos_325_erdos_325_2","label":"Erdős 325: erdos_325","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 325: erdos_325 erdosproblems/325.lean (fun x : n => (x : r) ^ (0.91709477 : r)) =o[attop]\n (fun x => (cardissumthreepowerbelow 3 x : r))"},{"id":"erdos_326_erdos_326_0","label":"Erdős 326: erdos_326","status":"open","category":"research","downstream_reach":2,"bridge_distance":1.0,"coverage_per_cost":0.3,"q":"erdos 326: erdos_326 erdosproblems/326.lean (a : n n), strictmono a \n minimal (fun a : set n a.isasymptoticaddbasisoforder 2) (set.range a) \n (c :"},{"id":"erdos_326_erdos_326_1","label":"Erdős 326: erdos_326","status":"proved","category":"research","downstream_reach":2,"bridge_distance":0.0,"coverage_per_cost":1.2,"q":"erdos 326: erdos_326 erdosproblems/326.lean (a : n n) (_ : strictmono a) (_ : set.range a |>.isaddbasisoforder 2) (x : r) (_ : 0 < x),\n tendsto (fun n (a"},{"id":"erdos_328_erdos_328_0","label":"Erdős 328: erdos_328","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 328: erdos_328 erdosproblems/328.lean c : n, 0 < c \n t : n, a : set n, ( n, sumrep a n c) \n p : fin t set n, ( i, p i) = a \n "},{"id":"erdos_329_erdos_329_0","label":"Erdős 329: erdos_329","status":"open","category":"research","downstream_reach":6,"bridge_distance":1.0,"coverage_per_cost":0.9,"q":"erdos 329: erdos_329 erdosproblems/329.lean ssup {sidonupperdensity a | (a : set n) (_ : issidon a)} =\n erdos_329.answer"},{"id":"erdos_329_erdos_329_1","label":"Erdős 329: erdos_329","status":"proved","category":"research","downstream_reach":6,"bridge_distance":0.0,"coverage_per_cost":3.6,"q":"erdos 329: erdos_329 erdosproblems/329.lean (a : set n), issidon a sidonupperdensity a 1/2"},{"id":"erdos_329_erdos_329_2","label":"Erdős 329: erdos_329","status":"proved","category":"research","downstream_reach":6,"bridge_distance":0.0,"coverage_per_cost":3.6,"q":"erdos 329: erdos_329 erdosproblems/329.lean (a : set n), issidon a \n sidonupperdensity a = 1 / real.sqrt 2"},{"id":"erdos_329_erdos_329_3","label":"Erdős 329: erdos_329","status":"proved","category":"research","downstream_reach":6,"bridge_distance":0.0,"coverage_per_cost":3.6,"q":"erdos 329: erdos_329 erdosproblems/329.lean (a : set n), issidon a sidonupperdensity a 1"},{"id":"erdos_329_erdos_329_4","label":"Erdős 329: erdos_329","status":"open","category":"research","downstream_reach":6,"bridge_distance":1.0,"coverage_per_cost":0.9,"q":"erdos 329: erdos_329 erdosproblems/329.lean (ssup {sidonupperdensity a | (a : set n) (_ : issidon a)} = 1) \n ( (a : finset n), issidon (a : set n) (d : set"},{"id":"erdos_329_exists_sidon_pos_density_5","label":"Erdős 329: exists_sidon_pos_density","status":"open","category":"textbook","downstream_reach":6,"bridge_distance":1.0,"coverage_per_cost":2.4,"q":"erdos 329: exists_sidon_pos_density erdosproblems/329.lean (a : set n), issidon a 0 < sidonupperdensity a"},{"id":"erdos_33_erdos_33_0","label":"Erdős 33: erdos_33","status":"open","category":"research","downstream_reach":3,"bridge_distance":1.0,"coverage_per_cost":0.45,"q":"erdos 33: erdos_33 erdosproblems/33.lean a : {a : set n | additivebasiscondition a}, filter.attop.limsup (fun n =>\n (a.1 icc 1 n).ncard / (n : ereal)) = "},{"id":"erdos_33_erdos_33_1","label":"Erdős 33: erdos_33","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 33: erdos_33 erdosproblems/33.lean 1 < a : {a : set n | additivebasiscondition a}, filter.attop.limsup (fun n =>\n (a.1 icc 1 n).ncard / (n : erea"},{"id":"erdos_33_erdos_33_2","label":"Erdős 33: erdos_33","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 33: erdos_33 erdosproblems/33.lean a : {a : set n | additivebasiscondition a}, filter.attop.limsup (fun n =>\n (a.1 icc 1 n).ncard / (n : ereal)) "},{"id":"erdos_330_erdos_330_statement_0","label":"Erdős 330: erdos_330_statement","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 330: erdos_330_statement erdosproblems/330.lean (a : set n), h, minasymptoticaddbasisoforder a h \n 0 < a.upperdensity n a, 0 < (unrepwithout a n h).upperd"},{"id":"erdos_331_erdos_331_0","label":"Erdős 331: erdos_331","status":"proved","category":"research","downstream_reach":2,"bridge_distance":0.0,"coverage_per_cost":1.2,"q":"erdos 331: erdos_331 erdosproblems/331.lean a b : set n,\n (fun (n : n) (n : r) ^ (1 / 2 : r)) =o[attop] (fun (n : n) (count ( a) n : r)) \n (fun (n"},{"id":"erdos_331_erdos_331_1","label":"Erdős 331: erdos_331","status":"proved","category":"research","downstream_reach":2,"bridge_distance":0.0,"coverage_per_cost":1.2,"q":"erdos 331: erdos_331 erdosproblems/331.lean a b : set n,\n ( c_a > 0, (fun (n : n) (count ( a) n : r)) ~[attop] (fun (n : n) c_a * (n : r) ^ (1 / 2 : r)"},{"id":"erdos_332_erdos_332_0","label":"Erdős 332: erdos_332","status":"open","category":"research","downstream_reach":1,"bridge_distance":2.0,"coverage_per_cost":0.1,"q":"erdos 332: erdos_332 erdosproblems/332.lean set n) : erdos_332.answer a hasboundedgaps (d_a a)"},{"id":"erdos_333_erdos_333_0","label":"Erdős 333: erdos_333","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 333: erdos_333 erdosproblems/333.lean a : set n, a.hasdensity 0 \n b : set n, a b + b \n (fun n => ((b set.icc 1 n).ncard : r)) =o[attop"},{"id":"erdos_337_erdos_337_0","label":"Erdős 337: erdos_337","status":"proved","category":"research","downstream_reach":4,"bridge_distance":0.0,"coverage_per_cost":2.4,"q":"erdos 337: erdos_337 erdosproblems/337.lean a : set n, a.isasymptoticaddbasis \n (fun n : n ((a icc 1 n).ncard : r)) =o[attop] (fun n : n (n : r)) \n"},{"id":"erdos_337_erdos_337_1","label":"Erdős 337: erdos_337","status":"proved","category":"research","downstream_reach":4,"bridge_distance":0.0,"coverage_per_cost":2.4,"q":"erdos 337: erdos_337 erdosproblems/337.lean h : n, 2 h \n a : set n, a.isasymptoticaddbasis \n (fun n : n ((a icc 1 n).ncard : r)) =o[attop] (fun "},{"id":"erdos_337_erdos_337_2","label":"Erdős 337: erdos_337","status":"proved","category":"research","downstream_reach":4,"bridge_distance":0.0,"coverage_per_cost":2.4,"q":"erdos 337: erdos_337 erdosproblems/337.lean a : set n, a.isasymptoticaddbasis \n (fun n : n ((a icc 1 n).ncard : r)) =o[attop] (fun n : n (n : r)) \n "},{"id":"erdos_337_erdos_337_3","label":"Erdős 337: erdos_337","status":"proved","category":"research","downstream_reach":4,"bridge_distance":0.0,"coverage_per_cost":2.4,"q":"erdos 337: erdos_337 erdosproblems/337.lean a : set n, a.isasymptoticaddbasis \n (fun n : n ((a icc 1 n).ncard : r)) =o[attop] (fun n : n (n : r)) \n "},{"id":"erdos_338_erdos_338_0","label":"Erdős 338: erdos_338","status":"open","category":"research","downstream_reach":12,"bridge_distance":1.0,"coverage_per_cost":1.8,"q":"erdos 338: erdos_338 erdosproblems/338.lean a : set n, ( h, hasorder a h) (( t, hasrestrictedorder a t) erdos_338.answer a)"},{"id":"erdos_338_erdos_338_1","label":"Erdős 338: erdos_338","status":"open","category":"research","downstream_reach":12,"bridge_distance":1.0,"coverage_per_cost":1.8,"q":"erdos 338: erdos_338 erdosproblems/338.lean f : n n, (a : set n) (h : n), hasorder a h \n ( t, hasrestrictedorder a t) hasrestrictedorder a (f h)"},{"id":"erdos_338_erdos_338_2","label":"Erdős 338: erdos_338","status":"open","category":"research","downstream_reach":12,"bridge_distance":1.0,"coverage_per_cost":1.8,"q":"erdos 338: erdos_338 erdosproblems/338.lean a : set n, ( h, hasorder a h) \n (restrictedorder a = basisorder a erdos_338.variants.eq_order.answer a)"},{"id":"erdos_338_erdos_338_3","label":"Erdős 338: erdos_338","status":"open","category":"research","downstream_reach":12,"bridge_distance":1.0,"coverage_per_cost":1.8,"q":"erdos 338: erdos_338 erdosproblems/338.lean a : set n, ( f : set n, f.finite h, hasorder (a \\ f) h) \n t, hasrestrictedorder a t"},{"id":"erdos_338_erdos_338_4","label":"Erdős 338: erdos_338","status":"open","category":"research","downstream_reach":12,"bridge_distance":1.0,"coverage_per_cost":1.8,"q":"erdos 338: erdos_338 erdosproblems/338.lean a : set n,\n ( h : n, f : set n, f.finite basisorder (a \\ f) = (h : n)) \n t, hasrestrictedorder a t"},{"id":"erdos_338_erdos_338_5","label":"Erdős 338: erdos_338","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 338: erdos_338 erdosproblems/338.lean n) (hh : 3 h) :\n basisorder (insert 1 {x : n | 0 < x h x}) = h \n t, hasrestrictedorder (insert 1 {x :"},{"id":"erdos_338_erdos_338_6","label":"Erdős 338: erdos_338","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 338: erdos_338 erdosproblems/338.lean set n) (ha : hasorder a 2) : hasrestrictedorder a 4"},{"id":"erdos_338_erdos_338_7","label":"Erdős 338: erdos_338","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 338: erdos_338 erdosproblems/338.lean set n) (ha : hasorder a 2)\n (hd : 0 < a.lowerdensity) : hasrestrictedorder a 3"},{"id":"erdos_338_erdos_338_8","label":"Erdős 338: erdos_338","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 338: erdos_338 erdosproblems/338.lean a : set n, hasorder a 2 hasrestrictedorder a 3"},{"id":"erdos_338_erdos_338_9","label":"Erdős 338: erdos_338","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 338: erdos_338 erdosproblems/338.lean basisorder (set.range fun k : n k ^ 2) = 4 \n restrictedorder (set.range fun k : n k ^ 2) = 5"},{"id":"erdos_338_erdos_338_10","label":"Erdős 338: erdos_338","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 338: erdos_338 erdosproblems/338.lean basisorder (set.range fun k : n k * (k + 1) / 2) = 3 \n restrictedorder (set.range fun k : n k * (k + 1) / 2) ="},{"id":"erdos_338_erdos_338_11","label":"Erdős 338: erdos_338","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 338: erdos_338 erdosproblems/338.lean k : n, 2 k a : set n,\n basisorder a = k restrictedorder a = (2 ^ (k - 2) + k - 1 : n)"},{"id":"erdos_34_erdos_34_0","label":"Erdős 34: erdos_34","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 34: erdos_34 erdosproblems/34.lean c : r, 0 < c n : n, n n, p : equiv.perm (fin n),\n ((consecutivesums n p).card : r) < c * (n : r) ^ 2"},{"id":"erdos_340_greedySidon_go_singleton_two_0","label":"Erdős 340: greedySidon_go_singleton_two","status":"proved","category":"test","downstream_reach":17,"bridge_distance":0.0,"coverage_per_cost":10.2,"q":"erdos 340: greedysidon_go_singleton_two erdosproblems/340.lean (greedysidon.go {1} (by simp [issidon]) 2).val = 2"},{"id":"erdos_340_greedySidon_go_pair_three_1","label":"Erdős 340: greedySidon_go_pair_three","status":"proved","category":"test","downstream_reach":17,"bridge_distance":0.0,"coverage_per_cost":10.2,"q":"erdos 340: greedysidon_go_pair_three erdosproblems/340.lean (greedysidon.go {1, 2} (by simp [issidon]) 3).val = 4"},{"id":"erdos_340_greedySidon_zero_2","label":"Erdős 340: greedySidon_zero","status":"proved","category":"test","downstream_reach":17,"bridge_distance":0.0,"coverage_per_cost":10.2,"q":"erdos 340: greedysidon_zero erdosproblems/340.lean greedysidon 0 = 1"},{"id":"erdos_340_greedySidon_one_3","label":"Erdős 340: greedySidon_one","status":"proved","category":"test","downstream_reach":17,"bridge_distance":0.0,"coverage_per_cost":10.2,"q":"erdos 340: greedysidon_one erdosproblems/340.lean greedysidon 1 = 2"},{"id":"erdos_340_greedySidon_two_4","label":"Erdős 340: greedySidon_two","status":"proved","category":"test","downstream_reach":17,"bridge_distance":0.0,"coverage_per_cost":10.2,"q":"erdos 340: greedysidon_two erdosproblems/340.lean greedysidon 2 = 4"},{"id":"erdos_340_greedySidon_three_5","label":"Erdős 340: greedySidon_three","status":"proved","category":"test","downstream_reach":17,"bridge_distance":0.0,"coverage_per_cost":10.2,"q":"erdos 340: greedysidon_three erdosproblems/340.lean greedysidon 3 = 8"},{"id":"erdos_340_greedySidon_four_6","label":"Erdős 340: greedySidon_four","status":"proved","category":"test","downstream_reach":17,"bridge_distance":0.0,"coverage_per_cost":10.2,"q":"erdos 340: greedysidon_four erdosproblems/340.lean greedysidon 4 = 13"},{"id":"erdos_340_greedySidon_five_7","label":"Erdős 340: greedySidon_five","status":"proved","category":"test","downstream_reach":17,"bridge_distance":0.0,"coverage_per_cost":10.2,"q":"erdos 340: greedysidon_five erdosproblems/340.lean greedysidon 5 = 21"},{"id":"erdos_340_greedySidon_ten_8","label":"Erdős 340: greedySidon_ten","status":"proved","category":"test","downstream_reach":17,"bridge_distance":0.0,"coverage_per_cost":10.2,"q":"erdos 340: greedysidon_ten erdosproblems/340.lean greedysidon 10 = 97"},{"id":"erdos_340_erdos_340_9","label":"Erdős 340: erdos_340","status":"open","category":"research","downstream_reach":17,"bridge_distance":1.0,"coverage_per_cost":2.55,"q":"erdos 340: erdos_340 erdosproblems/340.lean r) (h : > 0) :\n (fun n : n n / n ^ ) =o[attop]\n fun n : n ((set.range greedysidon set.icc 1 n).ncard "},{"id":"erdos_340_erdos_340_10","label":"Erdős 340: erdos_340","status":"open","category":"research","downstream_reach":17,"bridge_distance":1.0,"coverage_per_cost":2.55,"q":"erdos 340: erdos_340 erdosproblems/340.lean r) (h : > 0) :\n (fun n : n ((set.range greedysidon set.icc 1 n).ncard : r)) =[attop]\n erdos_340.variants"},{"id":"erdos_340_erdos_340_11","label":"Erdős 340: erdos_340","status":"open","category":"textbook","downstream_reach":17,"bridge_distance":1.0,"coverage_per_cost":6.8,"q":"erdos 340: erdos_340 erdosproblems/340.lean r) (h : > 0) :\n (fun n : n (n : r) ^ ((1 : r) / 3)) =o[attop]\n fun n : n ((set.range greedysidon set.ic"},{"id":"erdos_340_erdos_340_12","label":"Erdős 340: erdos_340","status":"open","category":"research","downstream_reach":17,"bridge_distance":1.0,"coverage_per_cost":2.55,"q":"erdos 340: erdos_340 erdosproblems/340.lean set.hasposdensity (set.range greedysidon - set.range greedysidon)"},{"id":"erdos_340_erdos_340_13","label":"Erdős 340: 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n, s.hasdensity 0 n sc, n set.range greedysidon - set.range greedysidon"},{"id":"erdos_341_erdos_341_0","label":"Erdős 341: erdos_341","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 341: erdos_341 erdosproblems/341.lean (a : n z),\n (f n in attop,\n isleast { x | a n < x x { a i + a j | (i n) (j n) } } (a (n + 1))"},{"id":"erdos_342_erdos_342_0","label":"Erdős 342: erdos_342","status":"proved","category":"test","downstream_reach":7,"bridge_distance":0.0,"coverage_per_cost":4.2,"q":"erdos 342: erdos_342 erdosproblems/342.lean a : n n, isulamsequence a a 0 = 1"},{"id":"erdos_342_erdos_342_1","label":"Erdős 342: erdos_342","status":"proved","category":"test","downstream_reach":7,"bridge_distance":0.0,"coverage_per_cost":4.2,"q":"erdos 342: erdos_342 erdosproblems/342.lean a : n n, isulamsequence a a 1 = 2"},{"id":"erdos_342_erdos_342_2","label":"Erdős 342: 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erdos_36.variants.upper.answer"},{"id":"erdos_36_erdos_36_16","label":"Erdős 36: erdos_36","status":"proved","category":"research","downstream_reach":18,"bridge_distance":0.0,"coverage_per_cost":10.8,"q":"erdos 36: erdos_36 erdosproblems/36.lean c, attop.tendsto minoverlapquotient (n c) c < 0.385694"},{"id":"erdos_36_erdos_36_17","label":"Erdős 36: erdos_36","status":"open","category":"research","downstream_reach":18,"bridge_distance":1.0,"coverage_per_cost":2.7,"q":"erdos 36: erdos_36 erdosproblems/36.lean attop.tendsto minoverlapquotient (n erdos_36.answer)"},{"id":"erdos_361_maxSubsetSumAvoidingCard_three_four_0","label":"Erdős 361: maxSubsetSumAvoidingCard_three_four","status":"open","category":"test","downstream_reach":2,"bridge_distance":2.0,"coverage_per_cost":0.2,"q":"erdos 361: maxsubsetsumavoidingcard_three_four erdosproblems/361.lean maxsubsetsumavoidingcard 3 4 = 2"},{"id":"erdos_361_erdos_361_1","label":"Erdős 361: 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n,\n (f n in attop, a a, 0 < a a < n (n - a).prime) \n liminf (fun n primedensityratio a n"},{"id":"erdos_429_erdos_429_0","label":"Erdős 429: erdos_429","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 429: erdos_429 erdosproblems/429.lean f : n n, tendsto f attop attop \n a : set n, a.infinite ( n, (a set.icc 1 n).ncard f n) \n ("},{"id":"erdos_43_erdos_43_0","label":"Erdős 43: erdos_43","status":"proved","category":"research","downstream_reach":2,"bridge_distance":0.0,"coverage_per_cost":1.2,"q":"erdos 43: erdos_43 erdosproblems/43.lean c : r, f n in filter.attop, (a b : finset n),\n a finset.icc 1 n \n b finset.icc 1 n \n issidon"},{"id":"erdos_43_erdos_43_1","label":"Erdős 43: erdos_43","status":"proved","category":"research","downstream_reach":2,"bridge_distance":0.0,"coverage_per_cost":1.2,"q":"erdos 43: erdos_43 erdosproblems/43.lean e (c > 0), o : n r, o =o[filter.attop] (1 : n r) \n f n in filter.attop, (a b : finset n),\n a fins"},{"id":"erdos_430_terms_eight_0","label":"Erdős 430: terms_eight","status":"proved","category":"test","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 430: terms_eight erdosproblems/430.lean terms 8 = {5, 7}"},{"id":"erdos_430_isGreatest_sub_one_1","label":"Erdős 430: isGreatest_sub_one","status":"proved","category":"API","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 430: isgreatest_sub_one erdosproblems/430.lean n} (hn : 3 n) : isgreatest (terms n : set n) (n - 1)"},{"id":"erdos_430_erdos_430_2","label":"Erdős 430: erdos_430","status":"open","category":"research","downstream_reach":3,"bridge_distance":1.0,"coverage_per_cost":0.45,"q":"erdos 430: erdos_430 erdosproblems/430.lean f n in attop, m terms n, m.prime"},{"id":"erdos_431_erdos_431_0","label":"Erdős 431: erdos_431","status":"proved","category":"research","downstream_reach":2,"bridge_distance":0.0,"coverage_per_cost":1.2,"q":"erdos 431: erdos_431 erdosproblems/431.lean a b : set n, a.infinite b.infinite \n (a + b : set n) =f[filter.cofinite] {n | n.prime}"},{"id":"erdos_431_erdos_431_1","label":"Erdős 431: erdos_431","status":"proved","category":"research","downstream_reach":2,"bridge_distance":0.0,"coverage_per_cost":1.2,"q":"erdos 431: erdos_431 erdosproblems/431.lean a b : set n, a.nontrivial b.nontrivial ((a + b) {n | n.prime}).infinite"},{"id":"erdos_433_erdos_433_0","label":"Erdős 433: erdos_433","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 433: erdos_433 erdosproblems/433.lean k, 2 k \n tendsto (fun n : n => (g k n : r) / ((n : r) ^ 2 / ((k : r) - 1))) attop (n 1)"},{"id":"erdos_434_erdos_434_0","label":"Erdős 434: erdos_434","status":"proved","category":"research","downstream_reach":2,"bridge_distance":0.0,"coverage_per_cost":1.2,"q":"erdos 434: erdos_434 erdosproblems/434.lean n) (hn : 1 n) (hk : 2 k) (h : k n) :\n isgreatest\n { nat.ncardunrepresentable s | (s : finset n) (_ : s f"},{"id":"erdos_434_erdos_434_1","label":"Erdős 434: erdos_434","status":"proved","category":"research","downstream_reach":2,"bridge_distance":0.0,"coverage_per_cost":1.2,"q":"erdos 434: erdos_434 erdosproblems/434.lean e (n 1) (k 2), k n \n isgreatest\n { nat.ncardunrepresentable s | (s : finset n) (_ : s finset.icc 1 n)\n"},{"id":"erdos_435_erdos_435_0","label":"Erdős 435: erdos_435","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 435: erdos_435 erdosproblems/435.lean n) (hn : n = 0)\n (hpk : p k : n, p.prime n = p ^ k) :\n isgreatest\n {m : z | c : n n, m = i fins"},{"id":"erdos_436_erdos_436_0","label":"Erdős 436: erdos_436","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 436: erdos_436 erdosproblems/436.lean k 2, k 2 < "},{"id":"erdos_436_erdos_436_1","label":"Erdős 436: erdos_436","status":"open","category":"research","downstream_reach":12,"bridge_distance":1.0,"coverage_per_cost":1.8,"q":"erdos 436: erdos_436 erdosproblems/436.lean k 3, odd k k 3 < "},{"id":"erdos_436_erdos_436_2","label":"Erdős 436: erdos_436","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 436: erdos_436 erdosproblems/436.lean 2 2 = 9"},{"id":"erdos_436_erdos_436_3","label":"Erdős 436: erdos_436","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 436: erdos_436 erdosproblems/436.lean 3 2 = 77"},{"id":"erdos_436_erdos_436_4","label":"Erdős 436: erdos_436","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 436: erdos_436 erdosproblems/436.lean 4 2 = 1224"},{"id":"erdos_436_erdos_436_5","label":"Erdős 436: erdos_436","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 436: erdos_436 erdosproblems/436.lean 5 2 = 7888"},{"id":"erdos_436_erdos_436_6","label":"Erdős 436: erdos_436","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 436: erdos_436 erdosproblems/436.lean 6 2 = 202124"},{"id":"erdos_436_erdos_436_7","label":"Erdős 436: erdos_436","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 436: erdos_436 erdosproblems/436.lean 7 2 = 1649375"},{"id":"erdos_436_erdos_436_8","label":"Erdős 436: erdos_436","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 436: erdos_436 erdosproblems/436.lean 3 3 = 23532"},{"id":"erdos_436_erdos_436_9","label":"Erdős 436: erdos_436","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 436: erdos_436 erdosproblems/436.lean k 2, even k k 3 = "},{"id":"erdos_436_erdos_436_10","label":"Erdős 436: erdos_436","status":"proved","category":"research","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 436: erdos_436 erdosproblems/436.lean k 2, l 4, k l = "},{"id":"erdos_436_r_two_two_thirteen_11","label":"Erdős 436: r_two_two_thirteen","status":"proved","category":"test","downstream_reach":12,"bridge_distance":0.0,"coverage_per_cost":7.2,"q":"erdos 436: r_two_two_thirteen erdosproblems/436.lean r 2 2 13 = 3"},{"id":"erdos_437_erdos_437_0","label":"Erdős 437: erdos_437","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 437: erdos_437 erdosproblems/437.lean : r, 0 < f x : n in attop, (x : r) ^ (1 - ) < l x"},{"id":"erdos_437_erdos_437_1","label":"Erdős 437: erdos_437","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 437: erdos_437 erdosproblems/437.lean (fun x : n (l x : r)) =o[attop] fun x : n (x : r)"},{"id":"erdos_437_erdos_437_2","label":"Erdős 437: erdos_437","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 437: erdos_437 erdosproblems/437.lean o1 o2 : n r, tendsto o1 attop (nhds 0) tendsto o2 attop (nhds 0) \n f x : n in attop,\n x * exp (-(2 "},{"id":"erdos_438_erdos_438_0","label":"Erdős 438: erdos_438","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 438: erdos_438 erdosproblems/438.lean tendsto (fun n : n (extremalsize n : r) / (n : r)) attop (n ((11 : r) / 32))"},{"id":"erdos_44_maxSidonSubsetCard_icc_bound_0","label":"Erdős 44: maxSidonSubsetCard_icc_bound","status":"proved","category":"textbook","downstream_reach":6,"bridge_distance":0.0,"coverage_per_cost":4.8,"q":"erdos 44: maxsidonsubsetcard_icc_bound erdosproblems/44.lean n) (hn : 1 n) :\n maxsidonsubsetcard (icc 1 n) 2 * real.sqrt n"},{"id":"erdos_44_erdos_44_1","label":"Erdős 44: erdos_44","status":"open","category":"research","downstream_reach":6,"bridge_distance":1.0,"coverage_per_cost":0.9,"q":"erdos 44: erdos_44 erdosproblems/44.lean e (n (1 : n)) (a finset.icc 1 n), issidon (a : set n) \n e ( > (0 : r)), e (m > n) (b finset.icc (n + 1) m)"},{"id":"erdos_44_erdos_44_2","label":"Erdős 44: erdos_44","status":"proved","category":"research","downstream_reach":6,"bridge_distance":0.0,"coverage_per_cost":3.6,"q":"erdos 44: erdos_44 erdosproblems/44.lean e ( > (0 : r)), f (m : n) in filter.attop,\n e (a finset.icc 1 m), issidon (a : set n) (1 - ) * real.sqrt m "},{"id":"erdos_44_example_sidon_set_3","label":"Erdős 44: example_sidon_set","status":"proved","category":"textbook","downstream_reach":6,"bridge_distance":0.0,"coverage_per_cost":4.8,"q":"erdos 44: example_sidon_set erdosproblems/44.lean issidon ({1, 2, 4, 8, 13} : set n)"},{"id":"erdos_44_sidon_set_lower_bound_4","label":"Erdős 44: sidon_set_lower_bound","status":"open","category":"textbook","downstream_reach":6,"bridge_distance":1.0,"coverage_per_cost":2.4,"q":"erdos 44: sidon_set_lower_bound erdosproblems/44.lean n) (hn : 1 n) :\n e (a finset.icc 1 n), issidon (a : set n) real.sqrt n / 2 a.card"},{"id":"erdos_44_greedy_sidon_construction_5","label":"Erdős 44: greedy_sidon_construction","status":"open","category":"textbook","downstream_reach":6,"bridge_distance":1.0,"coverage_per_cost":2.4,"q":"erdos 44: greedy_sidon_construction erdosproblems/44.lean n) (hn : 1 n) :\n e (a finset.icc 1 n), issidon (a : set n) n a.card ^ 3"},{"id":"erdos_440_erdos_440_0","label":"Erdős 440: erdos_440","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 440: erdos_440 erdosproblems/440.lean a : n n, strictmono a ( i, 0 < a i) \n (fun x : n (count a x : r)) =o[attop] fun x : n x"},{"id":"erdos_440_erdos_440_1","label":"Erdős 440: erdos_440","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 440: erdos_440 erdosproblems/440.lean isgreatest {l : r | a : n n, strictmono a ( i, 0 < a i) \n l = liminf (fun x : n (count a x : r) / x) att"},{"id":"erdos_440_erdos_440_2","label":"Erdős 440: erdos_440","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 440: erdos_440 erdosproblems/440.lean isgreatest {l : r | a : n n, strictmono a ( i, 0 < a i) \n l = limsup (fun x : n (count a x : r) / x) att"},{"id":"erdos_441_erdos_441_0","label":"Erdős 441: erdos_441","status":"proved","category":"research","downstream_reach":6,"bridge_distance":0.0,"coverage_per_cost":3.6,"q":"erdos 441: erdos_441 erdosproblems/441.lean n : n, 1 n g n = (erdosconstruction n).card"},{"id":"erdos_441_erdos_441_1","label":"Erdős 441: erdos_441","status":"proved","category":"research","downstream_reach":6,"bridge_distance":0.0,"coverage_per_cost":3.6,"q":"erdos 441: erdos_441 erdosproblems/441.lean f n : n in attop, (erdosconstruction n).card + real.iteratedlog n g n + 2"},{"id":"erdos_441_erdos_441_2","label":"Erdős 441: erdos_441","status":"proved","category":"research","downstream_reach":6,"bridge_distance":0.0,"coverage_per_cost":3.6,"q":"erdos 441: erdos_441 erdosproblems/441.lean c : r, n : n, (9 / 8 * n) - c g n"},{"id":"erdos_441_erdos_441_3","label":"Erdős 441: erdos_441","status":"proved","category":"research","downstream_reach":6,"bridge_distance":0.0,"coverage_per_cost":3.6,"q":"erdos 441: erdos_441 erdosproblems/441.lean c : r, n : n, (g n : r) (4 * n) + c"},{"id":"erdos_441_erdos_441_4","label":"Erdős 441: erdos_441","status":"proved","category":"research","downstream_reach":6,"bridge_distance":0.0,"coverage_per_cost":3.6,"q":"erdos 441: erdos_441 erdosproblems/441.lean (fun n : n (g n : r)) ~[attop] fun n (9 / 8 * n)"},{"id":"erdos_441_erdos_441_5","label":"Erdős 441: erdos_441","status":"proved","category":"research","downstream_reach":6,"bridge_distance":0.0,"coverage_per_cost":3.6,"q":"erdos 441: erdos_441 erdosproblems/441.lean (fun n : n (g n : r) - (9 / 8 * n)) =o[attop]\n fun n (n / real.log n) * real.log (real.log n)"},{"id":"erdos_442_bddProdUpper_0","label":"Erdős 442: bddProdUpper","status":"proved","category":"unknown","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 442: bddprodupper erdosproblems/442.lean set (n n)"},{"id":"erdos_442_erdos_442_1","label":"Erdős 442: erdos_442","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 442: erdos_442 erdosproblems/442.lean (a : set n),\n tendsto (fun (x : r) =>\n 1 / x.maxlogone.maxlogone * n (a icc 1 x+ : set n), (1 : r) /"},{"id":"erdos_442_erdos_442_2","label":"Erdős 442: erdos_442","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 442: erdos_442 erdosproblems/442.lean (a : set n) (f : r r) (c: r) (hc : 0 < c) (hf : f =o[attop] (1 : r r)),\n f (x : r) in attop,\n n ("},{"id":"erdos_443_erdos_443_0","label":"Erdős 443: erdos_443","status":"proved","category":"research","downstream_reach":2,"bridge_distance":0.0,"coverage_per_cost":1.2,"q":"erdos 443: erdos_443 erdosproblems/443.lean s : n, m n : n, n < m s (a n a m).card"},{"id":"erdos_443_erdos_443_1","label":"Erdős 443: erdos_443","status":"proved","category":"research","downstream_reach":2,"bridge_distance":0.0,"coverage_per_cost":1.2,"q":"erdos 443: erdos_443 erdosproblems/443.lean : r, 0 < n0 : n, m n : n, n0 < n n < m \n ((a n a m).card : r) < ((m : r) * n) ^ "},{"id":"erdos_444_divisorCount_zero_0","label":"Erdős 444: divisorCount_zero","status":"proved","category":"test","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 444: divisorcount_zero erdosproblems/444.lean set n) : divisorcount a 0 = 0"},{"id":"erdos_444_divisorCount_zero_two_three_four_twelve_1","label":"Erdős 444: divisorCount_zero_two_three_four_twelve","status":"proved","category":"test","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 444: divisorcount_zero_two_three_four_twelve erdosproblems/444.lean divisorcount {0, 2, 3, 4} 12 = 3"},{"id":"erdos_444_erdos_444_2","label":"Erdős 444: erdos_444","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 444: erdos_444 erdosproblems/444.lean a : set n, a.infinite k : n,\n attop.limsup (fun x : r (((ico 1 x+).sup (divisorcount a) /\n ( n ico "},{"id":"erdos_445_erdos_445_0","label":"Erdős 445: erdos_445","status":"open","category":"research","downstream_reach":4,"bridge_distance":1.0,"coverage_per_cost":0.6,"q":"erdos 445: erdos_445 erdosproblems/445.lean c : r, c > 1 / 2 \n f p : n in attop, p.prime n : n, erdos445prop c p n"},{"id":"erdos_445_erdos_445_1","label":"Erdős 445: erdos_445","status":"proved","category":"research","downstream_reach":4,"bridge_distance":0.0,"coverage_per_cost":2.4,"q":"erdos 445: erdos_445 erdosproblems/445.lean c0 < 1, c : r, c > c0 \n f p : n in attop, p.prime n : n, erdos445prop c p n"},{"id":"erdos_445_erdos_445_2","label":"Erdős 445: 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erdos_787","status":"open","category":"research","downstream_reach":9,"bridge_distance":1.0,"coverage_per_cost":1.35,"q":"erdos 787: erdos_787 erdosproblems/787.lean (fun n (g r n : r)) =[attop] erdos_787.answer"},{"id":"erdos_787_erdos_787_3","label":"Erdős 787: erdos_787","status":"proved","category":"research","downstream_reach":9,"bridge_distance":0.0,"coverage_per_cost":5.4,"q":"erdos 787: erdos_787 erdosproblems/787.lean n) : g z n = g r n"},{"id":"erdos_787_erdos_787_4","label":"Erdős 787: erdos_787","status":"proved","category":"research","downstream_reach":9,"bridge_distance":0.0,"coverage_per_cost":5.4,"q":"erdos 787: erdos_787 erdosproblems/787.lean c > (0 : r), f n : n in attop, c * real.log n (g r n : r)"},{"id":"erdos_787_erdos_787_5","label":"Erdős 787: erdos_787","status":"proved","category":"research","downstream_reach":9,"bridge_distance":0.0,"coverage_per_cost":5.4,"q":"erdos 787: erdos_787 erdosproblems/787.lean > (0 : r), f n : n in attop, (g r n : r) (n : r) ^ (2 / 5 + )"},{"id":"erdos_787_erdos_787_6","label":"Erdős 787: erdos_787","status":"proved","category":"research","downstream_reach":9,"bridge_distance":0.0,"coverage_per_cost":5.4,"q":"erdos 787: erdos_787 erdosproblems/787.lean c > (0 : r), f n : n in attop, real.log n ^ (1 + c) (g r n : r)"},{"id":"erdos_787_erdos_787_7","label":"Erdős 787: erdos_787","status":"proved","category":"research","downstream_reach":9,"bridge_distance":0.0,"coverage_per_cost":5.4,"q":"erdos 787: erdos_787 erdosproblems/787.lean c > (0 : r), f n : n in attop, (g r n : r) real.exp (c * real.sqrt (real.log n))"},{"id":"erdos_787_erdos_787_8","label":"Erdős 787: erdos_787","status":"proved","category":"research","downstream_reach":9,"bridge_distance":0.0,"coverage_per_cost":5.4,"q":"erdos 787: erdos_787 erdosproblems/787.lean c : r, 0 < c c < 1 / 68 f n : n in attop, real.log n ^ (1 + c) (g r n : r)"},{"id":"erdos_789_erdos_789_0","label":"Erdős 789: erdos_789","status":"open","category":"research","downstream_reach":8,"bridge_distance":1.0,"coverage_per_cost":1.2,"q":"erdos 789: erdos_789 erdosproblems/789.lean (fun n (subsetsumthreshold n : r)) =[attop] erdos_789.answer"},{"id":"erdos_789_erdos_789_1","label":"Erdős 789: erdos_789","status":"open","category":"research","downstream_reach":8,"bridge_distance":1.0,"coverage_per_cost":1.2,"q":"erdos 789: erdos_789 erdosproblems/789.lean (fun n (subsetsumthreshold n : r)) =[attop] fun n n"},{"id":"erdos_789_erdos_789_2","label":"Erdős 789: erdos_789","status":"proved","category":"research","downstream_reach":8,"bridge_distance":0.0,"coverage_per_cost":4.8,"q":"erdos 789: erdos_789 erdosproblems/789.lean (fun n (subsetsumthreshold n : r)) =o[attop] fun n n"},{"id":"erdos_789_erdos_789_3","label":"Erdős 789: erdos_789","status":"open","category":"research","downstream_reach":8,"bridge_distance":1.0,"coverage_per_cost":1.2,"q":"erdos 789: erdos_789 erdosproblems/789.lean (fun n : n n) =o[attop] fun n (subsetsumthreshold n : r)"},{"id":"erdos_789_erdos_789_4","label":"Erdős 789: erdos_789","status":"proved","category":"research","downstream_reach":8,"bridge_distance":0.0,"coverage_per_cost":4.8,"q":"erdos 789: erdos_789 erdosproblems/789.lean (fun n : n (n * real.log (real.log n) / real.log n)) =o[attop]\n fun n (subsetsumthreshold n : r)"},{"id":"erdos_789_erdos_789_5","label":"Erdős 789: erdos_789","status":"proved","category":"research","downstream_reach":8,"bridge_distance":0.0,"coverage_per_cost":4.8,"q":"erdos 789: erdos_789 erdosproblems/789.lean (fun n : n (n * real.log n) ^ ((1 : r) / 3)) =o[attop]\n fun n (subsetsumthreshold n : r)"},{"id":"erdos_789_erdos_789_6","label":"Erdős 789: erdos_789","status":"proved","category":"research","downstream_reach":8,"bridge_distance":0.0,"coverage_per_cost":4.8,"q":"erdos 789: erdos_789 erdosproblems/789.lean (fun n (subsetsumthreshold n : r)) =o[attop]\n fun n (n * real.log n) ^ ((1 : r) / 3)"},{"id":"erdos_789_erdos_789_7","label":"Erdős 789: erdos_789","status":"proved","category":"research","downstream_reach":8,"bridge_distance":0.0,"coverage_per_cost":4.8,"q":"erdos 789: erdos_789 erdosproblems/789.lean (fun n (subsetsumthreshold n : r)) =[attop]\n fun n (n * real.log n) ^ ((1 : r) / 3)"},{"id":"erdos_79_erdos_79_0","label":"Erdős 79: erdos_79","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 79: erdos_79 erdosproblems/79.lean (n : n), (n : n) (_ : n n) (g : simplegraph (fin n)),\n g.isramseysizelinear \n h : g.subgraph, h < "},{"id":"erdos_793_erdos_793_0","label":"Erdős 793: erdos_793","status":"proved","category":"research","downstream_reach":2,"bridge_distance":0.0,"coverage_per_cost":1.2,"q":"erdos 793: erdos_793 erdosproblems/793.lean c : r, tendsto (fun n : n ((f n : r) - nat.primecounting n) /\n ((n : r) ^ ((2 : r) / 3) / (log n) ^ 2)) attop ("},{"id":"erdos_793_erdos_793_1","label":"Erdős 793: erdos_793","status":"proved","category":"research","downstream_reach":2,"bridge_distance":0.0,"coverage_per_cost":1.2,"q":"erdos 793: erdos_793 erdosproblems/793.lean tendsto (fun n : n ((f n : r) - nat.primecounting n) /\n ((n : r) ^ ((2 : r) / 3) / (log n) ^ 2)) attop (n (27 / 2"},{"id":"erdos_794_erdos_794_0","label":"Erdős 794: erdos_794","status":"proved","category":"research","downstream_reach":4,"bridge_distance":0.0,"coverage_per_cost":2.4,"q":"erdos 794: erdos_794 erdosproblems/794.lean n : n, h : finset (finset (fin (3 * n))), h.isuniform 3 \n n ^ 3 + 1 h.card h.containssubgraph 4 3 h.co"},{"id":"erdos_794_erdos_794_1","label":"Erdős 794: erdos_794","status":"proved","category":"research","downstream_reach":4,"bridge_distance":0.0,"coverage_per_cost":2.4,"q":"erdos 794: erdos_794 erdosproblems/794.lean type*} [decidableeq v] (h : finset (finset v))\n (hh : h.isuniform 3) (h : h.containssubgraph 5 7) : h.containssubgrap"},{"id":"erdos_794_erdos_794_2","label":"Erdős 794: erdos_794","status":"proved","category":"research","downstream_reach":4,"bridge_distance":0.0,"coverage_per_cost":2.4,"q":"erdos 794: erdos_794 erdosproblems/794.lean harrishypergraph.isuniform 3 harrishypergraph.card = 28 \n harrishypergraph.containssubgraph 4 3 harrishype"},{"id":"erdos_794_erdos_794_3","label":"Erdős 794: erdos_794","status":"proved","category":"research","downstream_reach":4,"bridge_distance":0.0,"coverage_per_cost":2.4,"q":"erdos 794: erdos_794 erdosproblems/794.lean r) (h : 0 < ) :\n f n : n in attop, h : finset (finset (fin n)), h.isuniform 3 \n h.containssubgraph 4 3 "},{"id":"erdos_796_erdos_796_0","label":"Erdős 796: erdos_796","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 796: erdos_796 erdosproblems/796.lean c : r, tendsto normalizederror attop (n c)"},{"id":"erdos_798_erdos_798_0","label":"Erdős 798: erdos_798","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 798: erdos_798 erdosproblems/798.lean (fun n : n => (t n : r)) =o[attop] (fun n : n => (n : r))"},{"id":"erdos_798_erdos_798_1","label":"Erdős 798: erdos_798","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 798: erdos_798 erdosproblems/798.lean (fun n : n => (n : r) ^ (2 / 3 : r)) =o[attop] (fun n : n => (t n : r))"},{"id":"erdos_798_erdos_798_2","label":"Erdős 798: erdos_798","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 798: erdos_798 erdosproblems/798.lean (fun n : n => (t n : r)) =o[attop]\n (fun n : n => (n : r) ^ (2 / 3 : r) * real.log n)"},{"id":"erdos_80_erdos_80_0","label":"Erdős 80: erdos_80","status":"proved","category":"research","downstream_reach":5,"bridge_distance":0.0,"coverage_per_cost":3.0,"q":"erdos 80: erdos_80 erdosproblems/80.lean c : r, 0 < c c < 1 / 2 \n > (0 : r), f n : n in attop, (n : r) ^ < f c n"},{"id":"erdos_80_erdos_80_1","label":"Erdős 80: erdos_80","status":"open","category":"research","downstream_reach":5,"bridge_distance":1.0,"coverage_per_cost":0.75,"q":"erdos 80: erdos_80 erdosproblems/80.lean c : r, 0 < c c < 1 / 2 \n (fun n : n (f c n : r)) (fun n : n real.log n)"},{"id":"erdos_80_bookNumber_eq_zero_iff_2","label":"Erdős 80: bookNumber_eq_zero_iff","status":"proved","category":"API","downstream_reach":5,"bridge_distance":0.0,"coverage_per_cost":3.0,"q":"erdos 80: booknumber_eq_zero_iff erdosproblems/80.lean booknumber g = 0 e g.edgefinset, g.trianglescontaining e = "},{"id":"erdos_80_bookNumber_bot_3","label":"Erdős 80: bookNumber_bot","status":"proved","category":"test","downstream_reach":5,"bridge_distance":0.0,"coverage_per_cost":3.0,"q":"erdos 80: booknumber_bot erdosproblems/80.lean n} : booknumber ( : simplegraph (fin n)) = 0"},{"id":"erdos_80_everyEdgeInTriangle_bot_4","label":"Erdős 80: everyEdgeInTriangle_bot","status":"proved","category":"API","downstream_reach":5,"bridge_distance":0.0,"coverage_per_cost":3.0,"q":"erdos 80: everyedgeintriangle_bot erdosproblems/80.lean n} :\n everyedgeintriangle ( : simplegraph (fin n))"},{"id":"erdos_800_erdos_800_0","label":"Erdős 800: erdos_800","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 800: erdos_800 erdosproblems/800.lean c > (0 : r), (n : n) (v : type) [fintype v] (g : simplegraph v) [decidablerel g.adj],\n fintype.card v = n \n "},{"id":"erdos_801_erdos_801_0","label":"Erdős 801: erdos_801","status":"proved","category":"research","downstream_reach":1,"bridge_distance":0.0,"coverage_per_cost":0.6,"q":"erdos 801: erdos_801 erdosproblems/801.lean c : r, 0 < c f n : n in attop,\n g : simplegraph (fin n), (g.indepnum : r) n \n s : finset (fin n), ("},{"id":"erdos_802_erdos_802_0","label":"Erdős 802: erdos_802","status":"open","category":"research","downstream_reach":5,"bridge_distance":1.0,"coverage_per_cost":0.75,"q":"erdos 802: erdos_802 erdosproblems/802.lean r : n, c > (0 : r), n : n, g : simplegraph (fin n),\n g.cliquefree r \n c * (log g.averagedegree / g.aver"},{"id":"erdos_802_erdos_802_1","label":"Erdős 802: erdos_802","status":"proved","category":"research","downstream_reach":5,"bridge_distance":0.0,"coverage_per_cost":3.0,"q":"erdos 802: erdos_802 erdosproblems/802.lean r : n, c > (0 : r), n : n, g : simplegraph (fin n),\n g.cliquefree r \n c * (log (log (g.averagedegree + "},{"id":"erdos_802_erdos_802_2","label":"Erdős 802: erdos_802","status":"proved","category":"research","downstream_reach":5,"bridge_distance":0.0,"coverage_per_cost":3.0,"q":"erdos 802: erdos_802 erdosproblems/802.lean r : n, c > (0 : r), n : n, g : simplegraph (fin n),\n g.cliquefree r 2 g.averagedegree \n c * (log g."},{"id":"erdos_802_erdos_802_3","label":"Erdős 802: erdos_802","status":"proved","category":"research","downstream_reach":5,"bridge_distance":0.0,"coverage_per_cost":3.0,"q":"erdos 802: erdos_802 erdosproblems/802.lean c > (0 : r), n : n, g : simplegraph (fin n),\n g.cliquefree 3 \n c * (log g.averagedegree / g.averagedegree"},{"id":"erdos_802_erdos_802_4","label":"Erdős 802: erdos_802","status":"proved","category":"research","downstream_reach":5,"bridge_distance":0.0,"coverage_per_cost":3.0,"q":"erdos 802: erdos_802 erdosproblems/802.lean r : n, c > (0 : r), n : n, g : simplegraph (fin n),\n ( v, (g.induce (g.neighborset v)).colorable (r - 2)) \n"},{"id":"erdos_803_erdos_803_0","label":"Erdős 803: erdos_803","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 803: erdos_803 erdosproblems/803.lean (d c : r), 0 < c m 1, f n : n in attop, g : simplegraph (fin n),\n (n : r) * log n g.edgeset.ncard \n"},{"id":"erdos_803_erdos_803_1","label":"Erdős 803: erdos_803","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 803: erdos_803 erdosproblems/803.lean c : r, d : r, 1 < d f n : n in attop, g : simplegraph (fin n),\n (n : r) * log n g.edgeset.ncard \n "},{"id":"erdos_803_erdos_803_2","label":"Erdős 803: erdos_803","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 803: erdos_803 erdosproblems/803.lean d : r, k : n, c > 0, f n : n in attop, g : simplegraph (fin n),\n (n : r) * log n g.edgeset.ncard \n "},{"id":"erdos_804_erdos_804_0","label":"Erdős 804: erdos_804","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 804: erdos_804 erdosproblems/804.lean > 0, f n : n in attop,\n (n : r) ^ (1 / 2 - : r) f (log n) ^ 2+ n"},{"id":"erdos_804_erdos_804_1","label":"Erdős 804: erdos_804","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 804: erdos_804 erdosproblems/804.lean c > 0, f n : n in attop, c * (log n) ^ 3 f (log n) ^ 3+ n"},{"id":"erdos_804_erdos_804_2","label":"Erdős 804: erdos_804","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 804: erdos_804 erdosproblems/804.lean c2 c2 c3 c3 : r, 0 < c2 0 < c2 0 < c3 0 < c3 \n f n : n in attop,\n c2 * ((log n) ^ 2 / log (log n)) f"},{"id":"erdos_806_erdos_806_0","label":"Erdős 806: erdos_806","status":"proved","category":"research","downstream_reach":3,"bridge_distance":0.0,"coverage_per_cost":1.8,"q":"erdos 806: erdos_806 erdosproblems/806.lean : r, 0 < f n : n in attop,\n a : finset n, a finset.icc 1 n (a.card : r) n \n b : finset z, a"},{"id":"erdos_806_erdos_806_1","label":"Erdős 806: 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erdosproblems/825.lean r) (_hc : 0 < c)\n (h : (n : n) (_ : 1 n > c * n),\n s n.properdivisors, n = s.sum id) :\n 2 < c"},{"id":"erdos_826_erdos_826_0","label":"Erdős 826: erdos_826","status":"open","category":"research","downstream_reach":1,"bridge_distance":2.0,"coverage_per_cost":0.1,"q":"erdos 826: erdos_826 erdosproblems/826.lean c > (0 : r), { n | k 1, 0 (n + k) c * k }.infinite"},{"id":"erdos_828_erdos_828_0","label":"Erdős 828: erdos_828","status":"open","category":"research","downstream_reach":3,"bridge_distance":2.0,"coverage_per_cost":0.3,"q":"erdos 828: erdos_828 erdosproblems/828.lean a : z, set.infinite {n : n | ( n) n + a}"},{"id":"erdos_828_erdos_828_1","label":"Erdős 828: erdos_828","status":"open","category":"research","downstream_reach":3,"bridge_distance":2.0,"coverage_per_cost":0.3,"q":"erdos 828: erdos_828 erdosproblems/828.lean n > 1, n n - 1 prime n"},{"id":"erdos_828_erdos_828_2","label":"Erdős 828: 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950: erdos_950","status":"open","category":"research","downstream_reach":7,"bridge_distance":1.0,"coverage_per_cost":1.05,"q":"erdos 950: erdos_950 erdosproblems/950.lean f =o[attop] (fun n : n real.log (real.log n))"},{"id":"erdos_950_erdos_950_3","label":"Erdős 950: erdos_950","status":"proved","category":"research","downstream_reach":7,"bridge_distance":0.0,"coverage_per_cost":4.2,"q":"erdos 950: erdos_950 erdosproblems/950.lean (fun x : n n finset.range x, f n) ~[attop] (fun x : n (x : r)) \n (fun x : n n finset.range x, (f n) ^ "},{"id":"erdos_950_erdos_950_4","label":"Erdős 950: erdos_950","status":"open","category":"research","downstream_reach":7,"bridge_distance":1.0,"coverage_per_cost":1.05,"q":"erdos 950: erdos_950 erdosproblems/950.lean > 0, f x in attop, y < x,\n ( x : r) < ( y : r) + * ( (x - y) : r)"},{"id":"erdos_950_erdos_950_5","label":"Erdős 950: 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