ProblemP3014

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

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

Recorded solution requirements

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

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Record facts

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TheoremDB content license: CC0-1.0
Stable alias: erdos-problem-75-attempt-resolution-route
Source: www.erdosproblems.com ↗

Problem packetResearch packetR1259

R1259Recorded attemptOutcome: Open

Resolve the stated acceptance condition

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Authored summary

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

The author reports this result. The outcome applies to this attempt's recorded scope.

Attempt outcome: open strategy

Recorded scope: No scope is recorded.

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

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Authored record and scope

Authored title: Resolve the stated acceptance condition
Record type: attempt
Stored status: open_strategy
Evidence grade: self_reported

Work and source credit

Recorded action: No action description supplied.
Authored result summary: Read complete authored result summary
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 \(n^{1-\varepsilon}\)?
Reported outcome: No separate outcome supplied.
Recorded status: open_strategy
Recorded evidence grade: self_reported
Recorded scope: No explicit scope supplied.

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2Authored explanation

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.

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Agent handoff

Use this record for: 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 boundary: The recorded outcome applies to this attempt's stated scope.
Remaining questions: Unrecorded.
Blockers: Unrecorded.
Reopen when: Unrecorded.
Next useful step: Next test unrecorded. Inspect the argument, scope and related records.

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Replay material: source only

3Outcome

Replay package: source only

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Verification source: www.erdosproblems.com ↗, Editorial research route recorded 2026-07-31

4How it connects

Addresses

R1259addressesCurrent status and unresolved remainder
claim

Recorded for

R1259recorded forerdos problem 75
problem

Machine-readable record

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json

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  "body": "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.",
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6Provenance

View source, identifiers, and projection details

Project: erdos-problem-75-source-review
Locator: Editorial research route recorded 2026-07-31
License: CC0-1.0
Contributors: TheoremDB maintainers
Source: www.erdosproblems.com ↗
Public record: R1259
Stable alias: erdos-problem-75-attempt-resolution-route
Projection: Reproduction fields are derived from the immutable record.

A route someone took, recorded so the next person can reuse it or avoid it.

Resolve the stated acceptance condition

Work and source credit

2Authored explanation

3Outcome

4How it connects

Addresses

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6Provenance

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