[#P3012] Erdős's variant on almost-bipartite subgraphs of infinite-chromatic graphs

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Problem. 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 $b_G(n) \leq \sqrt{n}$?

A finite mathematical diagram showing an induced graph with edges marked for deletion toward bipartiteness. — An induced graph with a small set of edges marked for deletion.

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Definitions and notation

1Context

This is a specific variant of a problem posed by Erdős, Hajnal, and Szemerédi (1982) concerning how close to bipartite the finite subgraphs of an infinite-chromatic graph can be. The original problem asked whether for every function $f(n) \to \infty$, there exists a graph of infinite chromatic number whose $n$-vertex subgraphs can all be made bipartite by deleting at most $f(n)$ edges. This variant asks whether the specific bound $\sqrt{n}$ suffices.

2Definitions

Definition 1 (For a simple graph $A$, the edge distance to bipartite). For a simple graph $A$, the edge distance to bipartite is the minimum cardinality of a set of edges whose removal from $A$ yields a bipartite graph.

Definition 2 (For a simple graph $G$ and a natural number $n$, the maximum subgraph edge distance to bipartite $b_G(n)$). For a simple graph $G$ and a natural number $n$, the maximum subgraph edge distance to bipartite $b_G(n)$ is the maximum, over all induced subgraphs of $G$ with exactly $n$ vertices, of the edge distance to bipartite of that subgraph.

3What counts as a solution

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 $b_G(n) \leq \sqrt{n}$?

1Status

Saved packet · July 31, 2026

What counts as a solution

Saved packet status (Current status and unresolved remainder). OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 74 as open. The unresolved remainder is the full displayed statement. 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 $b_G(n) \leq \sqrt{n}$?^[1]

1Packet records

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Record	Kind	Assessment
OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 74 as open. The unresolved remainder is the full displayed statement. 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 $b_G(n) \leq \sqrt{n}$?	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 74 as open. The unresolved remainder is the full displayed statement. 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 $b_G(n) \leq \sqrt{n}$?^[1] Relevance to this problem For erdos problem 74, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 74 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 74: For. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 74 as open. The unresolved remainder is the full displayed statement. A complete resolution must satisfy this condition: 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 $b_G(n) \leq \sqrt{n}$? Open full record →
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 $b_G(n) \leq \sqrt{n}$?	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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 $b_G(n) \leq \sqrt{n}$?^[1] Relevance to this problem 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. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened 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. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 74 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 74 was open at commit 8138974387d9030542daabe67faaa33eff9356f8.
The pinned Formal Conjectures declaration was matched by problem number and source locator.
The controlled TheoremDB source corpus was checked for an already published record with the same slug.

How the 2 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemErdős's variant on almost-bipartite subgraphs of infinite-chromatic graphs

All 1 recorded relations between these records and the problem

Resolve the stated acceptance condition addresses Current status and unresolved remainder

2See also

Constructing a union from non-cancelling intersectionscombinatorics
Plane tilings by every five-cell lattice animalcombinatorics
Cycle Double Cover Conjecturecombinatorics

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Plain text

“Erdős's variant on almost-bipartite subgraphs of infinite-chromatic graphs.” TheoremDB. P3012. Problem statement; statement text SHA-256 2d4b7cd06badc22f6103f6bffe96e14426d6f8d6b4c847f071fc73a645c39711. https://theoremdb.org/statement/?ref=P3012

BibTeX

@misc{theoremdb-problem-2d4b7cd06badc22f6103f6bffe96e14426d6f8d6b4c847f071fc73a645c39711,
  title = {{Erdős's variant on almost-bipartite subgraphs of infinite-chromatic graphs}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 2d4b7cd06badc22f6103f6bffe96e14426d6f8d6b4c847f071fc73a645c39711},
  url = {https://theoremdb.org/statement/?ref=P3012}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 1 typed links, sourced from erdosproblems.com^[1], current as of July 31, 2026.

1References

Packet source. Erdős Problems database, Problem 74, maintained status record. Erdős Problems record 74, checked 2026-08-01. Problem 74; status field and linked bibliography. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Records the current open status and links the literature attached to this exact numbered problem.Also cited at Problem 74; status snapshot 8138974387d9030542daabe67faaa33eff9356f8.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.For Erdős's variant on almost-bipartite subgraphs of infinite-chromatic graphs: Records the current open status and links the literature attached to this exact numbered problem.Source named by the research packet.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 74 entry in data/problems.yaml. ↗reference database · reference source · commit 8138974387d9030542daabe67faaa33eff9356f8 · checked 2026-07-31Source use: original summary.Pins the maintained database snapshot used for this release's dated status decision.Source used to assess the problem's recorded status.For Erdős's variant on almost-bipartite subgraphs of infinite-chromatic graphs: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 74. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/74.lean:L132; theorem erdos_74.variants.sqrt; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. ↗reference database · primary source · commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 · checked 2026-07-31Source use: original summary.Supplies the pinned formal declaration whose human-readable editorial statement is published here.Source used to assess the problem's recorded status.For Erdős's variant on almost-bipartite subgraphs of infinite-chromatic graphs: Supplies the pinned formal declaration whose human-readable editorial statement is published here.

Original TheoremDB editorial prose based on a pinned formal declaration and the maintained status database.

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[#P3012] Erdős's variant on almost-bipartite subgraphs of infinite-chromatic graphs

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