{
  "h1": {
    "auc": 0.571,
    "ci_low": 0.511,
    "ci_high": 0.632,
    "threshold": 0.6,
    "supported": false
  },
  "h2": {
    "auc_obs": 0.571,
    "auc_obs_bridge": 0.567,
    "delta": -0.004,
    "delong_p": 0.54,
    "perm_p": 0.61,
    "supported": false
  },
  "gradient_boosting": {
    "auc": 0.49,
    "brier": 0.034,
    "base_rate_brier": 0.0341
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  "robustness": {
    "drop": {
      "prize": {
        "lr_obs_bridge_auc": 0.5591264942430391,
        "hgb_all_auc": 0.48190407509717165
      },
      "source_age": {
        "lr_obs_bridge_auc": 0.5665564573300412,
        "hgb_all_auc": 0.48559002737917717
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      "formalization": {
        "lr_obs_bridge_auc": 0.5789458405651844,
        "hgb_all_auc": 0.4910154435671157
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      "statement_len": {
        "lr_obs_bridge_auc": 0.5745013078446233,
        "hgb_all_auc": 0.4837214596523822
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      "status_edits": {
        "lr_obs_bridge_auc": 0.523781076588359,
        "hgb_all_auc": 0.49447450925269515
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      "oeis": {
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        "hgb_all_auc": 0.4760944031583836
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      "tags": {
        "lr_obs_bridge_auc": 0.5772606155425722,
        "hgb_all_auc": 0.49293290268169265
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      "prior_ai_attempts": {
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        "hgb_all_auc": 0.47186377025936876
      },
      "bridge_d_near": {
        "lr_obs_bridge_auc": 0.5681362602977486,
        "hgb_all_auc": 0.5041366697631212
      },
      "bridge_d_mass": {
        "lr_obs_bridge_auc": 0.5691996479820081,
        "hgb_all_auc": 0.49737972718605616
      },
      "bridge_d_2hop": {
        "lr_obs_bridge_auc": 0.567751240618965,
        "hgb_all_auc": 0.5008838646197472
      },
      "bridge_concept_cost": {
        "lr_obs_bridge_auc": 0.5657848901166059,
        "hgb_all_auc": 0.4906273681765957
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      "hub": {
        "lr_obs_bridge_auc": 0.5686328134549099,
        "hgb_all_auc": 0.49615362285183473
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  },
  "forecast_csv": {
    "problem_count": 593,
    "expected_3mo": 7.8,
    "model": "hgb_all",
    "repo_url": "https://github.com/Questboy23/ai-math-forecast",
    "sha256": "2c9a9d86ec22b205717723ce04ad9d8d6e013372f0f79b897e6813b3f1245981"
  },
  "scoring_dates": [
    {
      "label": "Scoring Date 1",
      "date": "2027-01-22T00:00:00-05:00"
    },
    {
      "label": "Scoring Date 2",
      "date": "2027-04-22T00:00:00-04:00"
    },
    {
      "label": "Scoring Date 3",
      "date": "2027-10-22T00:00:00-04:00"
    }
  ],
  "forecast_rows": [
    {
      "rank": 1,
      "number": 11,
      "p_ai_resolved_3mo": 0.083047,
      "p_ai_resolved_12mo": 0.29305,
      "statement": "Is every large odd integer $n$ the sum of a squarefree number and a power of 2?",
      "url": "https://www.erdosproblems.com/11",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 2,
      "number": 873,
      "p_ai_resolved_3mo": 0.080631,
      "p_ai_resolved_12mo": 0.285571,
      "statement": "Let $A=\\{a_1<a_2<\\cdots\\}\\subseteq \\mathbb{N}$ and let $F(A,X,k)$ count the number of $i$ such that\\[[a_i,a_{i+1},\\ldots,a_{i+k-1}] < X,\\]where the left-hand si",
      "url": "https://www.erdosproblems.com/873",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 3,
      "number": 1183,
      "p_ai_resolved_3mo": 0.078839,
      "p_ai_resolved_12mo": 0.279985,
      "statement": "Let $f(n)$ be maximal such that in any $2$-colouring of the subsets of $\\{1,\\ldots,n\\}$ there is always a monochromatic family of at least $f(n)$ sets which is ",
      "url": "https://www.erdosproblems.com/1183",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 4,
      "number": 75,
      "p_ai_resolved_3mo": 0.077041,
      "p_ai_resolved_12mo": 0.274346,
      "statement": "Is there a graph of chromatic number $\\aleph_1$ with $\\aleph_1$ vertices such that for all $\\epsilon>0$ if $n$ is sufficiently large and $H$ is a subgraph on $n",
      "url": "https://www.erdosproblems.com/75",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 5,
      "number": 408,
      "p_ai_resolved_3mo": 0.072951,
      "p_ai_resolved_12mo": 0.261397,
      "statement": "Let $\\phi(n)$ be the Euler totient function and $\\phi_k(n)$ be the iterated $\\phi$ function, so that $\\phi_1(n)=\\phi(n)$ and $\\phi_k(n)=\\phi(\\phi_{k-1}(n))$. Le",
      "url": "https://www.erdosproblems.com/408",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 6,
      "number": 1194,
      "p_ai_resolved_3mo": 0.072749,
      "p_ai_resolved_12mo": 0.260752,
      "statement": "Let $A\\subset\\mathbb{N}$ be such that every integer $n\\geq 1$ can be written uniquely as $a_n-b_n$ for some $a_n,b_n\\in A$. How fast must $a_n/n$ increase?",
      "url": "https://www.erdosproblems.com/1194",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 7,
      "number": 749,
      "p_ai_resolved_3mo": 0.069589,
      "p_ai_resolved_12mo": 0.250625,
      "statement": "Let $\\epsilon>0$. Does there exist $A\\subseteq \\mathbb{N}$ such that the lower density of $A+A$ is at least $1-\\epsilon$ and yet $1_A\\ast 1_A(n) \\ll_\\epsilon 1$",
      "url": "https://www.erdosproblems.com/749",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 8,
      "number": 52,
      "p_ai_resolved_3mo": 0.065061,
      "p_ai_resolved_12mo": 0.23593,
      "statement": "Let $A$ be a finite set of integers. Is it true that for every $\\epsilon>0$\\[\\max( \\lvert A+A\\rvert,\\lvert AA\\rvert)\\gg_\\epsilon \\lvert A\\rvert^{2-\\epsilon}?\\]",
      "url": "https://www.erdosproblems.com/52",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 9,
      "number": 595,
      "p_ai_resolved_3mo": 0.063524,
      "p_ai_resolved_12mo": 0.230892,
      "statement": "Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs?",
      "url": "https://www.erdosproblems.com/595",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 10,
      "number": 1188,
      "p_ai_resolved_3mo": 0.059992,
      "p_ai_resolved_12mo": 0.219225,
      "statement": "Call a set of distinct integers $1<n_1<\\cdots<n_k$ with associated congruence classes $a_i\\pmod{n_i}$ a distinct covering system if every integer satisfies at l",
      "url": "https://www.erdosproblems.com/1188",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 11,
      "number": 1143,
      "p_ai_resolved_3mo": 0.058146,
      "p_ai_resolved_12mo": 0.213074,
      "statement": "Let $p_1<\\cdots<p_u$ be primes and let $k\\geq 1$. Let $F_k(p_1,\\ldots,p_u)$ be such that every interval of $k$ positive integers contains at least $F_k(p_1,\\ldo",
      "url": "https://www.erdosproblems.com/1143",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 12,
      "number": 12,
      "p_ai_resolved_3mo": 0.057155,
      "p_ai_resolved_12mo": 0.209757,
      "statement": "Let $A$ be an infinite set such that there are no distinct $a,b,c\\in A$ such that $a\\mid (b+c)$ and $b,c>a$. Is there such an $A$ with\\[\\liminf \\frac{\\lvert A\\c",
      "url": "https://www.erdosproblems.com/12",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 13,
      "number": 1201,
      "p_ai_resolved_3mo": 0.056217,
      "p_ai_resolved_12mo": 0.206607,
      "statement": "Is it true that for every $\\epsilon,\\eta>0$ there exists a $k$ such that the density of $n$ for which\\[P(n(n+1)\\cdots(n+k))>n^{1-\\epsilon}\\]is at least $1-\\eta$",
      "url": "https://www.erdosproblems.com/1201",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 14,
      "number": 396,
      "p_ai_resolved_3mo": 0.056211,
      "p_ai_resolved_12mo": 0.206586,
      "statement": "Is it true that for every $k$ there exists $n$ such that\\[\\prod_{0\\leq i\\leq k}(n-i) \\mid \\binom{2n}{n}?\\]",
      "url": "https://www.erdosproblems.com/396",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 15,
      "number": 1209,
      "p_ai_resolved_3mo": 0.054217,
      "p_ai_resolved_12mo": 0.199861,
      "statement": "Let $A=\\{a_1<a_2<\\cdots\\}$ be a sequence of integers which tends to infinity sufficiently fast. If there is an $n$ such that all $n+a_k$ are primes then must th",
      "url": "https://www.erdosproblems.com/1209",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 16,
      "number": 1151,
      "p_ai_resolved_3mo": 0.052052,
      "p_ai_resolved_12mo": 0.192508,
      "statement": "Given $a_1,\\ldots,a_n\\in [-1,1]$ let\\[\\mathcal{L}^nf(x) = \\sum_{1\\leq i\\leq n}f(a_i)\\ell_i(x)\\]be the unique polynomial of degree $n-1$ which agrees with $f$ on",
      "url": "https://www.erdosproblems.com/1151",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 17,
      "number": 550,
      "p_ai_resolved_3mo": 0.046208,
      "p_ai_resolved_12mo": 0.172412,
      "statement": "Let $m_1\\leq\\cdots\\leq m_k$ and $n$ be sufficiently large. If $T$ is a tree on $n$ vertices and $G$ is the complete multipartite graph with vertex class sizes $",
      "url": "https://www.erdosproblems.com/550",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 18,
      "number": 1218,
      "p_ai_resolved_3mo": 0.045876,
      "p_ai_resolved_12mo": 0.171257,
      "statement": "Assume the Generalised Continuum Hypothesis and let $\\beta=\\omega_{\\omega+1}$. Is it true that\\[\\aleph_{\\beta+1} \\not\\to (\\aleph_\\beta,4,\\ldots,4)^3\\](where the",
      "url": "https://www.erdosproblems.com/1218",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 19,
      "number": 1207,
      "p_ai_resolved_3mo": 0.045549,
      "p_ai_resolved_12mo": 0.170123,
      "statement": "Let $P_d(n)$ be such that in any set of $n$ points in $\\mathbb{R}^d$ there exist at least $P_d(n)$ many points which do not contain an isosceles triangle. Estim",
      "url": "https://www.erdosproblems.com/1207",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 20,
      "number": 1206,
      "p_ai_resolved_3mo": 0.045273,
      "p_ai_resolved_12mo": 0.16916,
      "statement": "Does $\\{1,2^3,\\ldots,N^3\\}$ contain a Sidon set of size $\\gg N$? Is there an infinite set $A\\subset \\mathbb{N}$ of positive density such that $\\{a^3 : a\\in A\\}$",
      "url": "https://www.erdosproblems.com/1206",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 21,
      "number": 602,
      "p_ai_resolved_3mo": 0.045012,
      "p_ai_resolved_12mo": 0.168251,
      "statement": "Let $(A_i)$ be a family of sets with $\\lvert A_i\\rvert=\\aleph_0$ for all $i$, such that for any $i\\neq j$ we have $\\lvert A_i\\cap A_j\\rvert$ finite and $\\neq 1$",
      "url": "https://www.erdosproblems.com/602",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 22,
      "number": 1221,
      "p_ai_resolved_3mo": 0.044758,
      "p_ai_resolved_12mo": 0.167367,
      "statement": "Let $a=(a_1,\\ldots,a_n)$ be points, ordered consecutively, around the circle (normalised to have circumference $1$). Let\\[M_r(a)=\\max_r \\lvert a_{i+r}-a_i\\rvert",
      "url": "https://www.erdosproblems.com/1221",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 23,
      "number": 61,
      "p_ai_resolved_3mo": 0.0441,
      "p_ai_resolved_12mo": 0.16507,
      "statement": "For any graph $H$ is there some $c=c(H)>0$ such that every graph $G$ on $n$ vertices that does not contain $H$ as an induced subgraph contains either a complete",
      "url": "https://www.erdosproblems.com/61",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 24,
      "number": 1189,
      "p_ai_resolved_3mo": 0.043738,
      "p_ai_resolved_12mo": 0.163803,
      "statement": "Call a set of distinct integers $1<n_1<\\cdots<n_k$ a covering set if there is a choice of $a_i\\pmod{n_i}$ for $1\\leq i\\leq k$ such that every integer satisfies ",
      "url": "https://www.erdosproblems.com/1189",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 25,
      "number": 1192,
      "p_ai_resolved_3mo": 0.043214,
      "p_ai_resolved_12mo": 0.161971,
      "statement": "For $A\\subset \\mathbb{N}$ let $f_r(n)$ count the number of solutions to $n=a_1+\\cdots+a_r$ with $a_i\\in A$. Does there exist, for all $r\\geq 2$, a basis $A$ of ",
      "url": "https://www.erdosproblems.com/1192",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 26,
      "number": 524,
      "p_ai_resolved_3mo": 0.041779,
      "p_ai_resolved_12mo": 0.156933,
      "statement": "For any $t\\in (0,1)$ let $t=\\sum_{k=1}^\\infty \\epsilon_k(t)2^{-k}$ (where $\\epsilon_k(t)\\in \\{0,1\\}$). What is the correct order of magnitude (for almost all $t",
      "url": "https://www.erdosproblems.com/524",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 27,
      "number": 503,
      "p_ai_resolved_3mo": 0.038883,
      "p_ai_resolved_12mo": 0.146694,
      "statement": "What is the size of the largest $A\\subseteq \\mathbb{R}^d$ such that every three points from $A$ determine an isosceles triangle? That is, for any three points $",
      "url": "https://www.erdosproblems.com/503",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 28,
      "number": 462,
      "p_ai_resolved_3mo": 0.038042,
      "p_ai_resolved_12mo": 0.143701,
      "statement": "Let $p(n)$ denote the least prime factor of $n$. There is a constant $c>0$ such that\\[\\sum_{\\substack{n<x\\\\ n\\textrm{ not prime}}}\\frac{p(n)}{n}\\sim c\\frac{x^{1",
      "url": "https://www.erdosproblems.com/462",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 29,
      "number": 791,
      "p_ai_resolved_3mo": 0.036056,
      "p_ai_resolved_12mo": 0.13661,
      "statement": "Let $g(n)$ be minimal such that there exists $A\\subseteq \\{0,\\ldots,n\\}$ of size $g(n)$ with $\\{0,\\ldots,n\\}\\subseteq A+A$. Estimate $g(n)$. In particular is it",
      "url": "https://www.erdosproblems.com/791",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 30,
      "number": 160,
      "p_ai_resolved_3mo": 0.035799,
      "p_ai_resolved_12mo": 0.135689,
      "statement": "Let $h(N)$ be the smallest $k$ such that $\\{1,\\ldots,N\\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least ",
      "url": "https://www.erdosproblems.com/160",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 31,
      "number": 611,
      "p_ai_resolved_3mo": 0.035675,
      "p_ai_resolved_12mo": 0.135242,
      "statement": "For a graph $G$ let $\\tau(G)$ denote the minimal number of vertices that include at least one from each maximal clique of $G$ (sometimes called the clique trans",
      "url": "https://www.erdosproblems.com/611",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 32,
      "number": 872,
      "p_ai_resolved_3mo": 0.034188,
      "p_ai_resolved_12mo": 0.129898,
      "statement": "Consider the two-player game in which players alternately choose integers from $\\{2,3,\\ldots,n\\}$ to be included in some set $A$ (the same set for both players)",
      "url": "https://www.erdosproblems.com/872",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 33,
      "number": 555,
      "p_ai_resolved_3mo": 0.034017,
      "p_ai_resolved_12mo": 0.129281,
      "statement": "Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Determine the value of\\[R_k(C_{2",
      "url": "https://www.erdosproblems.com/555",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 34,
      "number": 306,
      "p_ai_resolved_3mo": 0.033905,
      "p_ai_resolved_12mo": 0.128876,
      "statement": "Let $a/b\\in \\mathbb{Q}_{>0}$ with $b$ squarefree. Are there integers $1<n_1<\\cdots<n_k$, each the product of two distinct primes, such that\\[\\frac{a}{b}=\\frac{1",
      "url": "https://www.erdosproblems.com/306",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 35,
      "number": 569,
      "p_ai_resolved_3mo": 0.033536,
      "p_ai_resolved_12mo": 0.127545,
      "statement": "Let $k\\geq 1$. What is the best possible $c_k$ such that\\[R(C_{2k+1},H)\\leq c_k m\\]for any graph $H$ on $m$ edges without isolated vertices?",
      "url": "https://www.erdosproblems.com/569",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 36,
      "number": 70,
      "p_ai_resolved_3mo": 0.032905,
      "p_ai_resolved_12mo": 0.125266,
      "statement": "Let $\\mathfrak{c}$ be the ordinal of the real numbers, $\\beta$ be any countable ordinal, and $2\\leq n<\\omega$. Is it true that $\\mathfrak{c}\\to (\\beta, n)_2^3$?",
      "url": "https://www.erdosproblems.com/70",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 37,
      "number": 1212,
      "p_ai_resolved_3mo": 0.032755,
      "p_ai_resolved_12mo": 0.124724,
      "statement": "Let $G$ be the graph with vertex set those pairs $(x,y)\\in \\mathbb{N}^2$ with $\\mathrm{gcd}(x,y)=1$, in which we join two vertices if the differ in only one coo",
      "url": "https://www.erdosproblems.com/1212",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 38,
      "number": 612,
      "p_ai_resolved_3mo": 0.032103,
      "p_ai_resolved_12mo": 0.12236,
      "statement": "Let $G$ be a connected graph with $n$ vertices, minimum degree $d$, and diameter $D$. Show if that $G$ contains no $K_{2r}$ and $(r-1)(3r+2)\\mid d$ then\\[D\\leq ",
      "url": "https://www.erdosproblems.com/612",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 39,
      "number": 415,
      "p_ai_resolved_3mo": 0.031862,
      "p_ai_resolved_12mo": 0.121484,
      "statement": "For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\\phi(m+1),\\ldots,\\phi(m+k)$ with $m+",
      "url": "https://www.erdosproblems.com/415",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 40,
      "number": 30,
      "p_ai_resolved_3mo": 0.031567,
      "p_ai_resolved_12mo": 0.120413,
      "statement": "Let $h(N)$ be the maximum size of a Sidon set in $\\{1,\\ldots,N\\}$. Is it true that, for every $\\epsilon>0$,\\[h(N) = N^{1/2}+O_\\epsilon(N^\\epsilon)?\\]",
      "url": "https://www.erdosproblems.com/30",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 41,
      "number": 857,
      "p_ai_resolved_3mo": 0.031233,
      "p_ai_resolved_12mo": 0.119201,
      "statement": "Let $m=m(n,k)$ be minimal such that in any collection of sets $A_1,\\ldots,A_m\\subseteq \\{1,\\ldots,n\\}$ there must exist a sunflower of size $k$ - that is, some ",
      "url": "https://www.erdosproblems.com/857",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 42,
      "number": 866,
      "p_ai_resolved_3mo": 0.031055,
      "p_ai_resolved_12mo": 0.118551,
      "statement": "Let $k\\geq 3$ and $g_k(N)$ be minimal such that if $A\\subseteq \\{1,\\ldots,2N\\}$ has $\\lvert A\\rvert \\geq N+g_k(N)$ then there exist integers $b_1,\\ldots,b_k$ su",
      "url": "https://www.erdosproblems.com/866",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 43,
      "number": 788,
      "p_ai_resolved_3mo": 0.030972,
      "p_ai_resolved_12mo": 0.118251,
      "statement": "Let $f(n)$ be maximal such that if $B\\subset (2n,4n)\\cap \\mathbb{N}$ there exists some $C\\subset (n,2n)\\cap \\mathbb{N}$ such that $c_1+c_2\\not\\in B$ for all $c_",
      "url": "https://www.erdosproblems.com/788",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 44,
      "number": 554,
      "p_ai_resolved_3mo": 0.030907,
      "p_ai_resolved_12mo": 0.118013,
      "statement": "Let $R_k(G)$ denote the minimal $m$ such that if the edges of $K_m$ are $k$-coloured then there is a monochromatic copy of $G$. Show that\\[\\lim_{k\\to \\infty}\\fr",
      "url": "https://www.erdosproblems.com/554",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 45,
      "number": 614,
      "p_ai_resolved_3mo": 0.03071,
      "p_ai_resolved_12mo": 0.117296,
      "statement": "Let $f(n,k)$ be minimal such that there is a graph with $n$ vertices and $f(n,k)$ edges where every set of $k+2$ vertices induces a subgraph with maximum degree",
      "url": "https://www.erdosproblems.com/614",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 46,
      "number": 1085,
      "p_ai_resolved_3mo": 0.030669,
      "p_ai_resolved_12mo": 0.117148,
      "statement": "Let $f_d(n)$ be minimal such that, in any set of $n$ points in $\\mathbb{R}^d$, there exist at most $f_d(n)$ pairs of points which distance $1$ apart. Estimate $",
      "url": "https://www.erdosproblems.com/1085",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 47,
      "number": 854,
      "p_ai_resolved_3mo": 0.030487,
      "p_ai_resolved_12mo": 0.116482,
      "statement": "Let $n_k$ denote the $k$th primorial, i.e. the product of the first $k$ primes. If $1=a_1<a_2<\\cdots a_{\\phi(n_k)}=n_k-1$ is the sequence of integers coprime to",
      "url": "https://www.erdosproblems.com/854",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 48,
      "number": 254,
      "p_ai_resolved_3mo": 0.029765,
      "p_ai_resolved_12mo": 0.113851,
      "statement": "Let $A\\subseteq \\mathbb{N}$ be such that\\[\\lvert A\\cap [1,2x]\\rvert -\\lvert A\\cap [1,x]\\rvert \\to \\infty\\textrm{ as }x\\to \\infty\\]and\\[\\sum_{n\\in A} \\{ \\theta n",
      "url": "https://www.erdosproblems.com/254",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 49,
      "number": 539,
      "p_ai_resolved_3mo": 0.029613,
      "p_ai_resolved_12mo": 0.113293,
      "statement": "Let $h(n)$ be such that, for any set $A\\subseteq \\mathbb{N}$ of size $n$, the set\\[\\left\\{ \\frac{a}{(a,b)}: a,b\\in A\\right\\}\\]has size at least $h(n)$. Estimate",
      "url": "https://www.erdosproblems.com/539",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 50,
      "number": 951,
      "p_ai_resolved_3mo": 0.029305,
      "p_ai_resolved_12mo": 0.112168,
      "statement": "Let $1<a_1<\\cdots$ be a sequence of real numbers such that\\[\\left\\lvert \\prod_i a_i^{k_i}-\\prod_j a_j^{\\ell_j}\\right\\rvert \\geq 1\\]for every distinct pair of no",
      "url": "https://www.erdosproblems.com/951",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 51,
      "number": 667,
      "p_ai_resolved_3mo": 0.029246,
      "p_ai_resolved_12mo": 0.111951,
      "statement": "Let $p,q\\geq 1$ be fixed integers. We define $H(n)=H(N;p,q)$ to be the largest $m$ such that any graph on $n$ vertices where every set of $p$ vertices spans at ",
      "url": "https://www.erdosproblems.com/667",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 52,
      "number": 670,
      "p_ai_resolved_3mo": 0.028979,
      "p_ai_resolved_12mo": 0.110975,
      "statement": "Let $A\\subseteq \\mathbb{R}^d$ be a set of $n$ points such that all pairwise distances differ by at least $1$. Is the diameter of $A$ at least $(1+o(1))n^2$?",
      "url": "https://www.erdosproblems.com/670",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 53,
      "number": 1203,
      "p_ai_resolved_3mo": 0.028933,
      "p_ai_resolved_12mo": 0.110806,
      "statement": "If $\\omega(n)$ counts the number of distinct prime divisors of $n$ then let\\[F(n)=\\max_k \\omega(n+k)\\frac{\\log\\log k}{\\log k}.\\]Prove that $F(n)\\to \\infty$ as $",
      "url": "https://www.erdosproblems.com/1203",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 54,
      "number": 1045,
      "p_ai_resolved_3mo": 0.028507,
      "p_ai_resolved_12mo": 0.109243,
      "statement": "Let $z_1,\\ldots,z_n\\in \\mathbb{C}$ with $\\lvert z_i-z_j\\rvert\\leq 2$ for all $i,j$, and\\[\\Delta(z_1,\\ldots,z_n)=\\prod_{i\\neq j}\\lvert z_i-z_j\\rvert.\\]What is th",
      "url": "https://www.erdosproblems.com/1045",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 55,
      "number": 1184,
      "p_ai_resolved_3mo": 0.028093,
      "p_ai_resolved_12mo": 0.107725,
      "statement": "Let $f(n,k)$ count the number of $1\\leq i\\leq k$ such that $P(n+i)>k$ (where $P(m)$ is the largest prime divisor of $m$). Is it true that, if $\\alpha>1$ is such",
      "url": "https://www.erdosproblems.com/1184",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 56,
      "number": 954,
      "p_ai_resolved_3mo": 0.028068,
      "p_ai_resolved_12mo": 0.107632,
      "statement": "Let $0=a_0<a_1<a_2<\\cdots$ be the sequence of integers defined by $a_0=0$ and $a_1=1$, and $a_{k+1}$ is the smallest integer $n$ for which the number of solutio",
      "url": "https://www.erdosproblems.com/954",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 57,
      "number": 686,
      "p_ai_resolved_3mo": 0.027115,
      "p_ai_resolved_12mo": 0.104127,
      "statement": "Can every integer $N\\geq 2$ be written as\\[N=\\frac{\\prod_{1\\leq i\\leq k}(m+i)}{\\prod_{1\\leq i\\leq k}(n+i)}\\]for some $k\\geq 2$ and $m\\geq n+k$?",
      "url": "https://www.erdosproblems.com/686",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 58,
      "number": 813,
      "p_ai_resolved_3mo": 0.026891,
      "p_ai_resolved_12mo": 0.103302,
      "statement": "Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle (a copy of $K_3$) must contain a clique on at le",
      "url": "https://www.erdosproblems.com/813",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 59,
      "number": 855,
      "p_ai_resolved_3mo": 0.026549,
      "p_ai_resolved_12mo": 0.102042,
      "statement": "If $\\pi(x)$ counts the number of primes in $[1,x]$ then is it true that (for large $x$ and $y$)\\[\\pi(x+y) \\leq \\pi(x)+\\pi(y)?\\]",
      "url": "https://www.erdosproblems.com/855",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 60,
      "number": 598,
      "p_ai_resolved_3mo": 0.026534,
      "p_ai_resolved_12mo": 0.101985,
      "statement": "Let $m$ be an infinite cardinal and $\\kappa$ be the successor cardinal of $2^{\\aleph_0}$. Can one colour the countable subsets of $m$ using $\\kappa$ many colour",
      "url": "https://www.erdosproblems.com/598",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 61,
      "number": 919,
      "p_ai_resolved_3mo": 0.02642,
      "p_ai_resolved_12mo": 0.101566,
      "statement": "Is there a graph $G$ with vertex set $\\omega_2^2$ and chromatic number $\\aleph_2$ such that every subgraph whose vertices have a lesser type has chromatic numbe",
      "url": "https://www.erdosproblems.com/919",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 62,
      "number": 787,
      "p_ai_resolved_3mo": 0.026375,
      "p_ai_resolved_12mo": 0.1014,
      "statement": "Let $g(n)$ be maximal such that given any set $A\\subset \\mathbb{R}$ with $\\lvert A\\rvert=n$ there exists some $B\\subseteq A$ of size $\\lvert B\\rvert\\geq g(n)$ s",
      "url": "https://www.erdosproblems.com/787",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 63,
      "number": 367,
      "p_ai_resolved_3mo": 0.026211,
      "p_ai_resolved_12mo": 0.100792,
      "statement": "Let $B_2(n)$ be the $2$-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once). Is it true that, for eve",
      "url": "https://www.erdosproblems.com/367",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 64,
      "number": 1199,
      "p_ai_resolved_3mo": 0.026113,
      "p_ai_resolved_12mo": 0.10043,
      "statement": "Is it true that in any $2$-colouring of $\\mathbb{N}$ there exists an infinite set $A$ such that all elements of $A+A$ are the same colour?",
      "url": "https://www.erdosproblems.com/1199",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 65,
      "number": 513,
      "p_ai_resolved_3mo": 0.025856,
      "p_ai_resolved_12mo": 0.09948,
      "statement": "Let $f=\\sum_{n=0}^\\infty a_nz^n$ be a transcendental entire function. What is the greatest possible value of\\[\\liminf_{r\\to \\infty} \\frac{\\max_n\\lvert a_nr^n\\rv",
      "url": "https://www.erdosproblems.com/513",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 66,
      "number": 1145,
      "p_ai_resolved_3mo": 0.025759,
      "p_ai_resolved_12mo": 0.099123,
      "statement": "Let $A=\\{1\\leq a_1<a_2<\\cdots\\}$ and $B=\\{1\\leq b_1<b_2<\\cdots\\}$ be sets of integers with $a_n/b_n\\to 1$. If $A+B$ contains all sufficiently large positive int",
      "url": "https://www.erdosproblems.com/1145",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 67,
      "number": 460,
      "p_ai_resolved_3mo": 0.025722,
      "p_ai_resolved_12mo": 0.098987,
      "statement": "Let $a_0=0$ and $a_1=1$, and in general define $a_k$ to be the least integer $>a_{k-1}$ for which $(n-a_k,n-a_i)=1$ for all $0\\leq i<k$. Does\\[\\sum_{0<a_i< n}\\f",
      "url": "https://www.erdosproblems.com/460",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 68,
      "number": 1132,
      "p_ai_resolved_3mo": 0.025573,
      "p_ai_resolved_12mo": 0.098436,
      "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let\\[l_k(x)=\\frac{\\prod_{i\\neq k}(x-x_i)}{\\prod_{i\\neq k}(x_k-x_i)},\\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\\ne",
      "url": "https://www.erdosproblems.com/1132",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 69,
      "number": 817,
      "p_ai_resolved_3mo": 0.025273,
      "p_ai_resolved_12mo": 0.097322,
      "statement": "Let $k\\geq 3$ and define $g_k(n)$ to be the minimal $N$ such that $\\{1,\\ldots,N\\}$ contains some $A$ of size $\\lvert A\\rvert=n$ such that\\[\\langle A\\rangle = \\l",
      "url": "https://www.erdosproblems.com/817",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 70,
      "number": 876,
      "p_ai_resolved_3mo": 0.02509,
      "p_ai_resolved_12mo": 0.096645,
      "statement": "Let $A=\\{a_1<a_2<\\cdots\\}\\subset \\mathbb{N}$ be an infinite sum-free set - that is, there are no solutions to\\[a=b_1+\\cdots+b_r\\]with $b_1<\\cdots<b_r<a\\in A$. H",
      "url": "https://www.erdosproblems.com/876",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 71,
      "number": 1200,
      "p_ai_resolved_3mo": 0.02469,
      "p_ai_resolved_12mo": 0.09516,
      "statement": "There exists a constant $C$ such that for all large $x$ there is a collection of primes $p_1<\\ldots<p_k<x$ with $\\sum\\frac{1}{p_i}<C$ together with a system of ",
      "url": "https://www.erdosproblems.com/1200",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 72,
      "number": 317,
      "p_ai_resolved_3mo": 0.024527,
      "p_ai_resolved_12mo": 0.094557,
      "statement": "Is there some constant $c>0$ such that for every $n\\geq 1$ there exists some $\\delta_k\\in \\{-1,0,1\\}$ for $1\\leq k\\leq n$ with\\[0< \\left\\lvert \\sum_{1\\leq k\\leq",
      "url": "https://www.erdosproblems.com/317",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 73,
      "number": 1097,
      "p_ai_resolved_3mo": 0.024126,
      "p_ai_resolved_12mo": 0.093068,
      "statement": "Let $A$ be a set of $n$ integers. How many distinct $d$ can occur as the common difference of a three-term arithmetic progression in $A$? In particular, are the",
      "url": "https://www.erdosproblems.com/1097",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 74,
      "number": 44,
      "p_ai_resolved_3mo": 0.0241,
      "p_ai_resolved_12mo": 0.092971,
      "statement": "Let $N\\geq 1$ and $A\\subset \\{1,\\ldots,N\\}$ be a Sidon set. Is it true that, for any $\\epsilon>0$, there exist $M$ and $B\\subset \\{N+1,\\ldots,M\\}$ (which may de",
      "url": "https://www.erdosproblems.com/44",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 75,
      "number": 1210,
      "p_ai_resolved_3mo": 0.024057,
      "p_ai_resolved_12mo": 0.09281,
      "statement": "Let $A\\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\\in A$. Is it true that\\[\\sum_{a\\in A}\\frac{1}{n-a}\\leq \\sum_{p<n}\\frac{1}{",
      "url": "https://www.erdosproblems.com/1210",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 76,
      "number": 875,
      "p_ai_resolved_3mo": 0.023688,
      "p_ai_resolved_12mo": 0.091438,
      "statement": "Let $A=\\{a_1<a_2<\\cdots\\}\\subset \\mathbb{N}$ be an infinite set such that the sets\\[S_r = \\{ a_1+\\cdots +a_r : a_1<\\cdots<a_r\\in A\\}\\]are disjoint for distinct ",
      "url": "https://www.erdosproblems.com/875",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 77,
      "number": 102,
      "p_ai_resolved_3mo": 0.023577,
      "p_ai_resolved_12mo": 0.091025,
      "statement": "Let $c>0$ and $h_c(n)$ be such that for any $n$ points in $\\mathbb{R}^2$ such that there are $\\geq cn^2$ lines each containing more than three points, there mus",
      "url": "https://www.erdosproblems.com/102",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 78,
      "number": 327,
      "p_ai_resolved_3mo": 0.023343,
      "p_ai_resolved_12mo": 0.090153,
      "statement": "Suppose $A\\subseteq \\{1,\\ldots,N\\}$ is such that if $a,b\\in A$ and $a\\neq b$ then $a+b\\nmid ab$. Can $A$ be 'substantially more' than the odd numbers? What if $",
      "url": "https://www.erdosproblems.com/327",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 79,
      "number": 161,
      "p_ai_resolved_3mo": 0.023247,
      "p_ai_resolved_12mo": 0.089796,
      "statement": "Let $\\alpha\\in[0,1/2)$ and $n,t\\geq 1$. Let $F^{(t)}(n,\\alpha)$ be the smallest $m$ such that we can $2$-colour the edges of the complete $t$-uniform hypergraph",
      "url": "https://www.erdosproblems.com/161",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 80,
      "number": 1170,
      "p_ai_resolved_3mo": 0.022967,
      "p_ai_resolved_12mo": 0.088752,
      "statement": "Is it consistent that\\[\\omega_2\\to (\\alpha)_2^2\\]for every $\\alpha <\\omega_2$?",
      "url": "https://www.erdosproblems.com/1170",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 81,
      "number": 390,
      "p_ai_resolved_3mo": 0.022858,
      "p_ai_resolved_12mo": 0.088344,
      "statement": "Let $f(n)$ be the minimal $m$ such that\\[n! = a_1\\cdots a_k\\]with $n< a_1<\\cdots <a_k=m$. Is there (and what is it) a constant $c$ such that\\[f(n)-2n \\sim c\\fra",
      "url": "https://www.erdosproblems.com/390",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 82,
      "number": 1131,
      "p_ai_resolved_3mo": 0.022807,
      "p_ai_resolved_12mo": 0.088156,
      "statement": "For $x_1,\\ldots,x_n\\in [-1,1]$ let\\[l_k(x)=\\frac{\\prod_{i\\neq k}(x-x_i)}{\\prod_{i\\neq k}(x_k-x_i)},\\]which are such that $l_k(x_k)=1$ and $l_k(x_i)=0$ for $i\\ne",
      "url": "https://www.erdosproblems.com/1131",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 83,
      "number": 200,
      "p_ai_resolved_3mo": 0.022433,
      "p_ai_resolved_12mo": 0.086756,
      "statement": "Does the longest arithmetic progression of primes in $\\{1,\\ldots,N\\}$ have length $o(\\log N)$?",
      "url": "https://www.erdosproblems.com/200",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 84,
      "number": 1122,
      "p_ai_resolved_3mo": 0.022359,
      "p_ai_resolved_12mo": 0.086479,
      "statement": "Let $f:\\mathbb{N}\\to \\mathbb{R}$ be an additive function (i.e. $f(ab)=f(a)+f(b)$ whenever $(a,b)=1$). Let\\[A=\\{ n \\geq 1: f(n+1)< f(n)\\}.\\]If $\\lvert A\\cap [1,X",
      "url": "https://www.erdosproblems.com/1122",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 85,
      "number": 1107,
      "p_ai_resolved_3mo": 0.02198,
      "p_ai_resolved_12mo": 0.085065,
      "statement": "Let $r\\geq 2$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\\mid n$. Is every large integer the sum of at most $r+1$ many $",
      "url": "https://www.erdosproblems.com/1107",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 86,
      "number": 839,
      "p_ai_resolved_3mo": 0.021754,
      "p_ai_resolved_12mo": 0.084216,
      "statement": "Let $1\\leq a_1<a_2<\\cdots$ be a sequence of integers such that no $a_i$ is the sum of consecutive $a_j$ for $j<i$. Is it true that\\[\\limsup \\frac{a_n}{n}=\\infty",
      "url": "https://www.erdosproblems.com/839",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 87,
      "number": 1016,
      "p_ai_resolved_3mo": 0.021728,
      "p_ai_resolved_12mo": 0.08412,
      "statement": "Let $h(n)$ be minimal such that there is a graph on $n$ vertices with $n+h(n)$ edges which contains a cycle on $k$ vertices, for all $3\\leq k\\leq n$. Estimate $",
      "url": "https://www.erdosproblems.com/1016",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 88,
      "number": 623,
      "p_ai_resolved_3mo": 0.021647,
      "p_ai_resolved_12mo": 0.083817,
      "statement": "Let $X$ be a set of cardinality $\\aleph_\\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\\not\\in A$ for all $A$. Must there ex",
      "url": "https://www.erdosproblems.com/623",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 89,
      "number": 187,
      "p_ai_resolved_3mo": 0.021533,
      "p_ai_resolved_12mo": 0.083391,
      "statement": "Find the best function $f(d)$ such that, in any 2-colouring of the integers, at least one colour class contains an arithmetic progression with common difference",
      "url": "https://www.erdosproblems.com/187",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 90,
      "number": 289,
      "p_ai_resolved_3mo": 0.021528,
      "p_ai_resolved_12mo": 0.083372,
      "statement": "Is it true that, for all sufficiently large $k$, there exist finite intervals $I_1,\\ldots,I_k\\subset \\mathbb{N}$, distinct, not overlapping or adjacent, with $\\",
      "url": "https://www.erdosproblems.com/289",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 91,
      "number": 942,
      "p_ai_resolved_3mo": 0.021465,
      "p_ai_resolved_12mo": 0.083135,
      "statement": "Let $h(n)$ count the number of powerful (if $p\\mid m$ then $p^2\\mid m$) integers in $[n^2,(n+1)^2)$. Estimate $h(n)$. In particular is there some constant $c>0$",
      "url": "https://www.erdosproblems.com/942",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 92,
      "number": 684,
      "p_ai_resolved_3mo": 0.021372,
      "p_ai_resolved_12mo": 0.082786,
      "statement": "For $0\\leq k\\leq n$ write\\[\\binom{n}{k} = uv\\]where the only primes dividing $u$ are in $[2,k]$ and the only primes dividing $v$ are in $(k,n]$. Let $f(n)$ be t",
      "url": "https://www.erdosproblems.com/684",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 93,
      "number": 489,
      "p_ai_resolved_3mo": 0.021136,
      "p_ai_resolved_12mo": 0.081902,
      "statement": "Let $A\\subseteq \\mathbb{N}$ be a set such that $\\lvert A\\cap [1,x]\\rvert=o(x^{1/2})$. Let\\[B=\\{ n\\geq 1 : a\\nmid n\\textrm{ for all }a\\in A\\}.\\]If $B=\\{b_1<b_2<\\",
      "url": "https://www.erdosproblems.com/489",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 94,
      "number": 486,
      "p_ai_resolved_3mo": 0.021094,
      "p_ai_resolved_12mo": 0.081742,
      "statement": "Let $A\\subseteq \\mathbb{N}$, and for each $n\\in A$ choose some $X_n\\subseteq \\mathbb{Z}/n\\mathbb{Z}$. Let\\[B = \\{ m\\in \\mathbb{N} : m\\not\\in X_n\\pmod{n}\\textrm{",
      "url": "https://www.erdosproblems.com/486",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 95,
      "number": 278,
      "p_ai_resolved_3mo": 0.02098,
      "p_ai_resolved_12mo": 0.081318,
      "statement": "Let $A=\\{n_1<\\cdots<n_r\\}$ be a finite set of positive integers. What is the maximum density of integers covered by a suitable choice of congruences $a_i\\pmod{n",
      "url": "https://www.erdosproblems.com/278",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 96,
      "number": 170,
      "p_ai_resolved_3mo": 0.020862,
      "p_ai_resolved_12mo": 0.080871,
      "statement": "Let $F(N)$ be the smallest possible size of $A\\subset \\{0,1,\\ldots,N\\}$ such that $\\{0,1,\\ldots,N\\}\\subset A-A$. Find the value of\\[\\lim_{N\\to \\infty}\\frac{F(N)",
      "url": "https://www.erdosproblems.com/170",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 97,
      "number": 1101,
      "p_ai_resolved_3mo": 0.020683,
      "p_ai_resolved_12mo": 0.080202,
      "statement": "If $u=\\{u_1<u_2<\\cdots\\}$ is a sequence of integers such that $(u_i,u_j)=1$ for all $i\\neq j$ and $\\sum \\frac{1}{u_i}<\\infty$ then let $\\{a_1<a_2<\\cdots\\}$ be t",
      "url": "https://www.erdosproblems.com/1101",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 98,
      "number": 887,
      "p_ai_resolved_3mo": 0.020651,
      "p_ai_resolved_12mo": 0.08008,
      "statement": "Is there an absolute constant $K$ such that, for every $C>0$, if $n$ is sufficiently large then $n$ has at most $K$ divisors in $(n^{1/2},n^{1/2}+C n^{1/4})$.",
      "url": "https://www.erdosproblems.com/887",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 99,
      "number": 940,
      "p_ai_resolved_3mo": 0.020291,
      "p_ai_resolved_12mo": 0.078728,
      "statement": "Let $r\\geq 3$. A number $n$ is $r$-powerful if for every prime $p$ which divides $n$ we have $p^r\\mid n$. Are there infinitely many integers which are not the s",
      "url": "https://www.erdosproblems.com/940",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    },
    {
      "rank": 100,
      "number": 820,
      "p_ai_resolved_3mo": 0.020112,
      "p_ai_resolved_12mo": 0.078052,
      "statement": "Let $H(n)$ be the smallest integer $l$ such that there exist $k<l$ with $(k^n-1,l^n-1)=1$. Is it true that $H(n)=3$ infinitely often? (That is, $(2^n-1,3^n-1)=1",
      "url": "https://www.erdosproblems.com/820",
      "forecast_date": "2026-10-22",
      "model": "hgb_all"
    }
  ]
}