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

- ID: `P3034`
- Reference: `erdos-problem-108`
- Page: https://theoremdb.org/statements/P3034
- Record maturity: Reviewed problem with recorded work

## 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 $k$?

### Context

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 $G$.

### Problem setup

- **Definition (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 (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 (A simple graph).** A simple graph is an undirected graph without loops or multiple edges.
- **Definition (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.** 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 $G$.

### What 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 $k$?

## 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. 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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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 $k$?

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 $k$?

### Background and intake notes

- 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.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-108`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Erdős Problems database, Problem 108, maintained status record. Erdős Problems record 108, checked 2026-08-01. Problem 108; status field and linked bibliography https://www.erdosproblems.com/108
   - 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
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source use: original_summary
   - Records the current open status and links the literature attached to this exact numbered problem.
   - 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.
2. <a id="reference-2"></a>Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 108 entry in data/problems.yaml https://github.com/teorth/erdosproblems/blob/8138974387d9030542daabe67faaa33eff9356f8/data/problems.yaml
   - reference_database; reference source; commit 8138974387d9030542daabe67faaa33eff9356f8; checked 2026-07-31
   - Source 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.
3. <a id="reference-3"></a>Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 108. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/108.lean:L36; theorem erdos_108; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/108.lean#L36
   - reference_database; primary source; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1; checked 2026-07-31
   - Source 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.