{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1257",
  "content_hash": null,
  "slug": "erdos-problem-74-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 74: For a simple graph $G$ and a natural number $n$, let the edge distance to bipartite of a subgraph $A$ of $G$ be the minimum number of edges that must be deleted from $A$ to obtain a bipartite graph. Let the maximum subgraph edge distance to bipartite of $G$ at $n$, denoted $b_G(n)$, be the supremum of the edge distances to bipartite over all induced subgraphs of $G$ on exactly $n$ vertices (this supremum is finite because deleting all edges of an $n$-vertex graph, at most $\\binom{n}{2}$, always yields a bipartite graph). Does there exist a graph $G$ with infinite chromatic number such that for every natural number $n$, we have $b_G(n) \\leq \\sqrt{n}$?",
  "relevance": "For Erdős's variant on almost-bipartite subgraphs of infinite-chromatic graphs, record erdos-problem-74-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 74: For a simple graph $G$ and a natural number $n$, let the edge distance to bipartite of a subgraph $A$ of $G$ be the minimum number of edges that must be deleted from $A$ to obtain a bipartite graph.",
  "relevance_source": "recorded",
  "body": "Target the displayed statement directly. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 74: For a simple graph $G$ and a natural number $n$, let the edge distance to bipartite of a subgraph $A$ of $G$ be the minimum number of edges that must be deleted from $A$ to obtain a bipartite graph. Let the maximum subgraph edge distance to bipartite of $G$ at $n$, denoted $b_G(n)$, be the supremum of the edge distances to bipartite over all induced subgraphs of $G$ on exactly $n$ vertices (this supremum is finite because deleting all edges of an $n$-vertex graph, at most $\\binom{n}{2}$, always yields a bipartite graph). Does there exist a graph $G$ with infinite chromatic number such that for every natural number $n$, we have $b_G(n) \\leq \\sqrt{n}$? 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/74",
      "locator": "Editorial research route recorded 2026-07-31"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.erdosproblems.com/74",
    "locator": "Editorial research route recorded 2026-07-31"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1258",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "addresses",
      "direction": "outgoing"
    },
    {
      "slug": "erdos-problem-74",
      "title": "erdos problem 74",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}