{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1249",
  "content_hash": null,
  "slug": "erdos-problem-61-attempt-resolution-route",
  "type": "attempt",
  "title": "Resolve the stated acceptance condition",
  "summary": "Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 61: For a finite simple graph $H$, say that a simple graph $G$ is $H$\\textit{-free} if $H$ does not appear as an induced subgraph of $G$. For a graph $G$, let $\\alpha(G)$ denote the independence number of $G$ (the size of a largest independent set), and let $\\omega(G)$ denote the clique number of $G$ (the size of a largest clique). For a function $f : \\mathbb{N} \\to \\mathbb{R}$, say that $f$ is an \\textit{Erdős–Hajnal lower bound} for $H$ if for all sufficiently large $n$, every $H$-free graph $G$ on $n$ vertices satisfies $\\alpha(G) \\geq f(n)$ or $\\omega(G) \\geq f(n)$. Does there exist, for every finite simple graph $H$, a constant $c > 0$ such that the function $n \\mapsto n^{c}$ is an Erdős–Hajnal lower bound for $H$?",
  "relevance": "For The Erdős–Hajnal Conjecture on Polynomial Ramsey-Type Bounds, record erdos-problem-61-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 61: For a finite simple graph $H$, say that a simple graph $G$ is $H$\\textit{-free} if $H$ does not appear as an induced subgraph of $G$.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 61: For a finite simple graph $H$, say that a simple graph $G$ is $H$\\textit{-free} if $H$ does not appear as an induced subgraph of $G$. For a graph $G$, let $\\alpha(G)$ denote the independence number of $G$ (the size of a largest independent set), and let $\\omega(G)$ denote the clique number of $G$ (the size of a largest clique). For a function $f : \\mathbb{N} \\to \\mathbb{R}$, say that $f$ is an \\textit{Erdős–Hajnal lower bound} for $H$ if for all sufficiently large $n$, every $H$-free graph $G$ on $n$ vertices satisfies $\\alpha(G) \\geq f(n)$ or $\\omega(G) \\geq f(n)$. Does there exist, for every finite simple graph $H$, a constant $c > 0$ such that the function $n \\mapsto n^{c}$ is an Erdős–Hajnal lower bound for $H$? Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress.",
  "status": "open_strategy",
  "evidence_grade": "self_reported",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://www.erdosproblems.com/61",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/61",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1250",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-61",
      "title": "erdos problem 61",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}