LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Every smaller positive exponent is again an Erdős–Hajnal constant

Statement

Let $C$ be a hereditary graph class. If $ϵ$ is an Erdős–Hajnal constant for $C$ and $0 < δ \leq ϵ$ , then $δ$ is also an Erdős–Hajnal constant for $C$ .

Facts & Assumptions

Given: A hereditary class $C$ , an Erdős–Hajnal constant $ϵ$ for it, and a real $δ$ with $0 < δ \leq ϵ$ .

[L1]

A positive real $c$ is an Erdős–Hajnal constant for $C$ exactly when every nonempty $G \in C$ satisfies $hom (G) \geq ∣ V (G) ∣^{c}$ , with $a^{c} = \exp (c \log a)$ for $a > 0$ (The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class).

[L2]

The logarithm is strictly increasing and $\log 1 = 0$ (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

[L3]

The exponential function is strictly increasing on $R$ (The exponential function is strictly increasing).

Proof

technique · direct

1.1

Let $G \in C$ be nonempty and put $n = ∣ V (G) ∣ \geq 1$ .

given

1.2

If $n = 1$ , then $n^{δ} = n^{ϵ} = 1$ , so the required inequality follows from the one for $ϵ$ .

L1algebra

1.3

If $n > 1$ , then $\log n > 0$ by [L2], and hence $δ \log n \leq ϵ \log n$ .

givenL2algebra

2.1

In the case $n > 1$ , [L3] and the real-power convention in [L1] give $n^{δ} = \exp (δ \log n) \leq \exp (ϵ \log n) = n^{ϵ}$ .

step 1.3L1L3

3.1

In both cases, $hom (G) \geq n^{ϵ} \geq n^{δ}$ ; since $G$ was arbitrary, $δ$ is an Erdős–Hajnal constant for $C$ .

step 1.2step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 49 results over 12 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Erdos-Hajnal properties in graphs and hypergraphs, introduction (standard reference, not scraped)