ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-05

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The Lemma 3.5 parameter choice at the next power of two above the source threshold

Example

Consider the special case in which the constant supplied by the source lemma happens to be $c_{1} = 1$ . Then $q = 42$ and the threshold is $m = 2^{42}$ . Let $∣ G ∣ : = 2^{43}$ , and define

x : = ∣ G ∣^{- 1 / 3} = 2^{- 43 / 3}, ϵ : = x^{1 / 7} = 2^{- 43 / 21} .

Then the size thresholds in the source Erdős-Hajnal reduction are

∣ F ∣_{\min} : = ϵ^{2} ∣ G ∣, ∣ S ∣_{\min} : = ϵ^{3} ∣ G ∣ .

Facts & Assumptions

Given: The numerical choices displayed in the Example.

[A1]

Conditionally on the displayed value $c_{1} = 1$ , one has $∣ G ∣ = 2^{43} > 2^{42} = m$ , and $2^{43}$ is the first power of two strictly above the corresponding source threshold. The example does not assert that the existential constant $c_{1}$ can be freely chosen.

Verification

technique · direct arithmetic

1.1

Since $x = 2^{- 43 / 3}$ , one has $ϵ = x^{1 / 7} = 2^{- 43 / 21} > 2^{- 43 / 3} = x .$ Also $ϵ = 2^{- 43 / 21} < 2^{- 2} = \frac{1}{4},$ because $43 / 21 > 2$ . Thus $x < ϵ \leq \frac{1}{4}$ .

A1algebra

2.1

The first threshold is $∣ F ∣_{\min} = ϵ^{2} ∣ G ∣ = 2^{- 86 / 21} \cdot 2^{43} = 2^{817 / 21},$ while $ϵ^{- 1} = 2^{43 / 21} .$ Since $817 / 21 > 43 / 21$ , one gets $∣ F ∣_{\min} > ϵ^{- 1}$ .

step 1.1algebra

2.2

The second threshold is $∣ S ∣_{\min} = ϵ^{3} ∣ G ∣ = 2^{- 129 / 21} \cdot 2^{43} = 2^{774 / 21} .$ Again $774 / 21 > 43 / 21$ , so $∣ S ∣_{\min} > ϵ^{- 1}$ .

step 1.1algebra

3.1

Therefore, in the special case $c_{1} = 1$ , this concrete instance satisfies all of the numerical inequalities used in the source Erdős-Hajnal reduction at the next power-of-two graph order above the threshold.

step 2.1step 2.2∎

Depends on

Leaf-reducible wonderful generalized nice finite families have the Erdős-Hajnal property

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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

Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Lemma 3.5 (standard reference, not scraped)