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

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

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 $k$?

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 $G$.

2Problem setup

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.

Remark 1. 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 $G$.

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

Current 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

Record	Kind	Assessment
	Result	Supported
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 →
	Trace	Reported
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 →
No records match these filters.

Notes and companion materialContext, examples, and computations

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 connectTyped relations and evidence flow

ProblemErdős's problem on high-girth, high-chromatic-number subgraphs

Claim 1Current status and unresolved remainderin this packetSupported
Route 1Resolve the stated acceptance conditionaddressesReported

2See also

Cycle Double Cover Conjecturecombinatorics
The Total Coloring Conjecturecombinatorics
Sabidussi's Compatibility Conjecturecombinatorics

How to cite

TheoremDB contributors, “Erdős's problem on high-girth, high-chromatic-number subgraphs,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/erdos-problem-108

This page as plain text: erdos-problem-108.md

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 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

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Report a problem

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance