[#P3034] Erdős's problem on high-girth, high-chromatic-number subgraphs

Problem. Erdős Problem 108: For every integer $r \geq 4$ and every integer $k \geq 2$, does there exist a finite integer $f(k,r)$ such that every nonempty simple graph $G$ with chromatic number at least $f(k,r)$ contains a subgraph $H$ whose girth is at least $r$ and whose chromatic number is at least $k$?

A finite mathematical diagram showing a colored graph with a sparse high-girth subgraph marked. — A colored graph with a sparse subgraph marked.

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Definitions and notation

1Context

This problem asks whether graphs with sufficiently large chromatic number must contain subgraphs that simultaneously have large girth and large chromatic number. The girth of a graph is the length of its shortest cycle. The chromatic number of a graph is the smallest number of colors needed to color its vertices so that no two adjacent vertices share the same color. A subgraph of a graph $G$ is a graph whose vertex set and edge set are subsets of those of $G$.

2Definitions

Definition 1 (The girth of a graph). The girth of a graph is the length of its shortest cycle, or infinity if the graph contains no cycles.

Definition 2 (The chromatic number of a graph). The chromatic number of a graph is the minimum number of colors required to assign a color to each vertex such that no two adjacent vertices receive the same color.

Definition 3 (A simple graph). A simple graph is an undirected graph without loops or multiple edges.

Definition 4 (A subgraph of a graph $G$). A subgraph of a graph $G$ is a graph formed from a subset of the vertices of $G$ and a subset of the edges of $G$ that connect vertices in the chosen subset.

3What counts as a solution

Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 108: For every integer $r \geq 4$ and every integer $k \geq 2$, does there exist a finite integer $f(k,r)$ such that every nonempty simple graph $G$ with chromatic number at least $f(k,r)$ contains a subgraph $H$ whose girth is at least $r$ and whose chromatic number is at least $k$?

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 108 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 108: For every integer $r \geq 4$ and every integer $k \geq 2$, does there exist a finite integer $f(k,r)$ such that every nonempty simple graph $G$ with chromatic number at least $f(k,r)$ contains a subgraph $H$ whose girth is at least $r$ and whose chromatic number is at least $k$?^[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 108 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 108: For every integer $r \geq 4$ and every integer $k \geq 2$, does there exist a finite integer $f(k,r)$ such that every nonempty simple graph $G$ with chromatic number at least $f(k,r)$ contains a subgraph $H$ whose girth is at least $r$ and whose chromatic number is at least $k$?	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 108 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 108: For every integer $r \geq 4$ and every integer $k \geq 2$, does there exist a finite integer $f(k,r)$ such that every nonempty simple graph $G$ with chromatic number at least $f(k,r)$ contains a subgraph $H$ whose girth is at least $r$ and whose chromatic number is at least $k$?^[1] Relevance to this problem For erdos problem 108, pins the dated research frontier: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 108 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 108. 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 108 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 108: For every integer $r \geq 4$ and every integer $k \geq 2$, does there exist a finite integer $f(k,r)$ such that every nonempty simple graph $G$ with chromatic number at least $f(k,r)$ contains a subgraph $H$ whose girth is at least $r$ and whose chromatic number is at least $k$? Open full record →
Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 108: For every integer $r \geq 4$ and every integer $k \geq 2$, does there exist a finite integer $f(k,r)$ such that every nonempty simple graph $G$ with chromatic number at least $f(k,r)$ contains a subgraph $H$ whose girth is at least $r$ and whose chromatic number is at least $k$?	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 108: For every integer $r \geq 4$ and every integer $k \geq 2$, does there exist a finite integer $f(k,r)$ such that every nonempty simple graph $G$ with chromatic number at least $f(k,r)$ contains a subgraph $H$ whose girth is at least $r$ and whose chromatic number is at least $k$?^[1] Relevance to this problem For Erdős's problem on high-girth, high-chromatic-number subgraphs, record erdos-problem-108-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 108: For every integer $r \geq 4$ and every integer $k \geq 2$, does there exist a finite integer $f(k,r)$ such that every nonempty simple graph $G$ with chromatic number at least $f(k,r)$ contains a subgraph $H$ whose girth is at least $r$ and whose chromatic number is at least $k$? 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 108: For every integer $r \geq 4$ and every integer $k \geq 2$, does there exist a finite integer $f(k,r)$ such that every nonempty simple graph $G$ with chromatic number at least $f(k,r)$ contains a subgraph $H$ whose girth is at least $r$ and whose chromatic number is at least $k$? 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 108 as open. The unresolved remainder is the full displayed statement.

The maintained database entry for Erdős Problem 108 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 problem on high-girth, high-chromatic-number subgraphs

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 problem on high-girth, high-chromatic-number subgraphs.” TheoremDB. P3034. Problem statement; statement text SHA-256 f5d1ff7ab01808918195d43c33b873bb39f1c3ada35e9f8a81413424e7ffa6f2. https://theoremdb.org/statement/?ref=P3034

BibTeX

@misc{theoremdb-problem-f5d1ff7ab01808918195d43c33b873bb39f1c3ada35e9f8a81413424e7ffa6f2,
  title = {{Erdős's problem on high-girth, high-chromatic-number subgraphs}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 f5d1ff7ab01808918195d43c33b873bb39f1c3ada35e9f8a81413424e7ffa6f2},
  url = {https://theoremdb.org/statement/?ref=P3034}
}

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.

1Lean verification

1References

Packet source. Erdős Problems database, Problem 108, maintained status record. Erdős Problems record 108, checked 2026-08-01. Problem 108; 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 108; 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 problem on high-girth, high-chromatic-number subgraphs: 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 108 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 problem on high-girth, high-chromatic-number subgraphs: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 108. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/108.lean:L36; theorem erdos_108; 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 problem on high-girth, high-chromatic-number subgraphs: 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.

[#P3034] Erdős's problem on high-girth, high-chromatic-number subgraphs

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