# P3014: Erdős's problem on uncountable graphs with large independent sets in all large finite subgraphs

- ID: `P3014`
- Reference: `erdos-problem-75`
- Page: https://theoremdb.org/statements/P3014
- Record maturity: Reviewed problem with recorded work

## Problem

Erdős Problem 75: Does there exist a graph with chromatic number \(\aleph_1\) and exactly \(\aleph_1\) vertices such that for every real number \(\varepsilon > 0\), there exists a natural number \(N\) with the following property: for every natural number \(n \geq N\) and every subgraph \(H\) on exactly \(n\) vertices, there exists an independent set \(I\) contained in the vertex set of \(H\) whose cardinality exceeds \(n^{1-\varepsilon}\)?

### Context

This problem asks whether an uncountable graph can have chromatic number \(\aleph_1\) while simultaneously having the property that all sufficiently large finite subgraphs contain independent sets that are nearly as large as the subgraph itself. The condition that the independent set size exceeds \(n^{1-\varepsilon}\) for all \(\varepsilon > 0\) means that the independence ratio of large finite subgraphs decays slower than any polynomial rate.

### Problem setup

- **Definition (The chromatic number of a graph).** The chromatic number of a graph is the smallest cardinal \(\kappa\) such that the vertices can be partitioned into \(\kappa\) independent sets, where an independent set is a set of vertices with no edges between any two of them.
- **Definition (A subgraph of a graph \(G\).** A subgraph of a graph \(G\) is a graph whose vertex set is a subset of the vertex set of \(G\) and whose edge set is a subset of the edge set of \(G\) restricted to that vertex subset.
- **Definition (The notation \(\aleph_1\).** The notation \(\aleph_1\) denotes the first uncountable cardinal, which is the cardinality of the set of all countable ordinal numbers.
- **Definition (The notation \(\forall^\infty n \in \mathbb{N}\), or equivalently the filter condition \(\forall^\mathcal{F} n \to \infty\).** The notation \(\forall^\infty n \in \mathbb{N}\), or equivalently the filter condition \(\forall^\mathcal{F} n \to \infty\), means that there exists some natural number \(N\) such that the stated property holds for all \(n \geq N\).
- **Remark.** This problem asks whether an uncountable graph can have chromatic number \(\aleph_1\) while simultaneously having the property that all sufficiently large finite subgraphs contain independent sets that are nearly as large as the subgraph itself. The condition that the independent set size exceeds \(n^{1-\varepsilon}\) for all \(\varepsilon > 0\) means that the independence ratio of large finite subgraphs decays slower than any polynomial rate.

### What counts as a solution

- Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 75: Does there exist a graph with chromatic number \(\aleph_1\) and exactly \(\aleph_1\) vertices such that for every real number \(\varepsilon > 0\), there exists a natural number \(N\) with the following property: for every natural number \(n \geq N\) and every subgraph \(H\) on exactly \(n\) vertices, there exists an independent set \(I\) contained in the vertex set of \(H\) whose cardinality exceeds \(n^{1-\varepsilon}\)?

## Status

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 75 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 75: Does there exist a graph with chromatic number \(\aleph_1\) and exactly \(\aleph_1\) vertices such that for every real number \(\varepsilon > 0\), there exists a natural number \(N\) with the following property: for every natural number \(n \geq N\) and every subgraph \(H\) on exactly \(n\) vertices, there exists an independent set \(I\) contained in the vertex set of \(H\) whose cardinality exceeds \(n^{1-\varepsilon}\)? [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 75 as open. The unresolved remainder is the full displayed statement. Give a rigorous proof or counterexample that completely resolves this statement: Erdős Problem 75: Does there exist a graph with chromatic number \(\aleph_1\) and exactly \(\aleph_1\) vertices such that for every real number \(\varepsilon > 0\), there exists a natural number \(N\) with the following property: for every natural number \(n \geq N\) and every subgraph \(H\) on exactly \(n\) vertices, there exists an independent set \(I\) contained in the vertex set of \(H\) whose cardinality exceeds \(n^{1-\varepsilon}\)?

OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 75 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 75: Does there exist a graph with chromatic number \(\aleph_1\) and exactly \(\aleph_1\) vertices such that for every real number \(\varepsilon > 0\), there exists a natural number \(N\) with the following property: for every natural number \(n \geq N\) and every subgraph \(H\) on exactly \(n\) vertices, there exists an independent set \(I\) contained in the vertex set of \(H\) whose cardinality exceeds \(n^{1-\varepsilon}\)?

### Background and intake notes

- Original intake status: OPEN as of 2026-07-31. The maintained Erdős Problems database at commit 8138974387d9030542daabe67faaa33eff9356f8 lists Problem 75 as open. The unresolved remainder is the full displayed statement.
- The maintained database entry for Erdős Problem 75 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 75: Does there exist a graph with chromatic number \(\aleph_1\) and exactly \(\aleph_1\) vertices such that for every real number \(\varepsilon > 0\), there exists a natural number \(N\) with the following property: for every natural number \(n \geq N\) and every subgraph \(H\) on exactly \(n\) vertices, there exists an independent set \(I\) contained in the vertex set of \(H\) whose cardinality exceeds \(n^{1-\varepsilon}\)? [1](#reference-1)

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `erdos-problem-75`, 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 75, maintained status record. Erdős Problems record 75, checked 2026-08-01. Problem 75; status field and linked bibliography https://www.erdosproblems.com/75
   - Also cited at Problem 75; 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 uncountable graphs with large independent sets in all large finite 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 75 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 uncountable graphs with large independent sets in all large finite 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 75. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/75.lean:L36; theorem erdos_75; commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1 https://github.com/google-deepmind/formal-conjectures/blob/bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1/FormalConjectures/ErdosProblems/75.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 uncountable graphs with large independent sets in all large finite subgraphs: Supplies the pinned formal declaration whose human-readable editorial statement is published here.