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

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 \(n^{1-\varepsilon}\)?

A finite mathematical diagram showing a graph with a large independent vertex set marked. — A finite graph with an independent vertex set marked.

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1Context

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.

2Definitions

Definition 1 (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 2 (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 3 (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 4 (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 \(n \geq N\).

3What 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 \(n^{1-\varepsilon}\)?

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 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]

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 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 \(n^{1-\varepsilon}\)?	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 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] Relevance to this problem For erdos problem 75, pins the dated research frontier: 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. 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 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 \(n^{1-\varepsilon}\)? Open full record →
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 \(n^{1-\varepsilon}\)?	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 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] Relevance to this problem For Erdős's problem on uncountable graphs with large independent sets in all large finite subgraphs, record erdos-problem-75-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 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 \(n^{1-\varepsilon}\)? 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 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 \(n^{1-\varepsilon}\)? 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 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.

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 uncountable graphs with large independent sets in all large finite 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 uncountable graphs with large independent sets in all large finite subgraphs.” TheoremDB. P3014. Problem statement; statement text SHA-256 c2c5f9488ee3c2c7d265fa1374ebaa882d252dbdab32ad794722fc0211efabec. https://theoremdb.org/statement/?ref=P3014

BibTeX

@misc{theoremdb-problem-c2c5f9488ee3c2c7d265fa1374ebaa882d252dbdab32ad794722fc0211efabec,
  title = {{Erdős's problem on uncountable graphs with large independent sets in all large finite subgraphs}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 c2c5f9488ee3c2c7d265fa1374ebaa882d252dbdab32ad794722fc0211efabec},
  url = {https://theoremdb.org/statement/?ref=P3014}
}

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 75, maintained status record. Erdős Problems record 75, checked 2026-08-01. Problem 75; 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 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.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.
Erdős problem database, data/problems.yaml, commit 8138974387d9030542daabe67faaa33eff9356f8. Problem 75 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 uncountable graphs with large independent sets in all large finite subgraphs: Pins the maintained database snapshot used for this release's dated status decision.
Google DeepMind, Formal Conjectures, formal statement for Erdős Problem 75. GitHub commit bb1c69dab14ad895ac1ccdb7b0c0e0a67c8eb3e1. FormalConjectures/ErdosProblems/75.lean:L36; theorem erdos_75; 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 uncountable graphs with large independent sets in all large finite 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.

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

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