CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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The universal logarithmic Ramsey guarantee cannot be replaced by any universal positive power

Statement refuted

There is a universal exponent $ϵ > 0$ such that every nonempty finite graph $G$ satisfies $hom (G) \geq ∣ V (G) ∣^{ϵ}$ .

Facts & Assumptions

Given: An arbitrary real exponent $ϵ > 0$ .

[L1]

For every $n \geq 16$ , some $n$ -vertex graph $G$ satisfies $hom (G) < 3 \log_{2} n$ (For every $n \geq 16$ there is an $n$ -vertex graph with $hom (G) < 3 \log_{2} n$ ).

[L2]

For every $a > 0$ , $\log x / x^{a} \to 0$ as $x \to + \infty$ (The logarithm grows more slowly than every positive real power).

[L3]

For $x > 0$ , $\log_{2} x = \log x / \log 2$ (Change of base and inversion of the positive-base real exponential).

Counterexample

technique · constructive

1.1givenL2L3choose

By [L2] and [L3], choose an integer $n \geq 16$ with $3 \log_{2} n < n^{ϵ}$ .

2.1step 1.1L1chooseconstruct

By [L1], choose an $n$ -vertex graph $G$ with $hom (G) < 3 \log_{2} n < n^{ϵ}$ .

3.1givenstep 2.1discharge-construct∎

Thus every proposed positive universal exponent has a finite counterexample, so the statement is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

A. Chernikov, MATH 223M notes, sec. 3.1 (standard reference, not scraped)