LemmaStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-08-26 (gpt-5.4)

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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

1.1given

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

1.2L1algebra

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

1.3givenL2algebra

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

2.1step 1.3L1L3

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

3.1step 1.2step 2.1L1∎

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

Depends on

Used by

The singleton Bird family has property (*) Corollary
The singleton E-graph family is generalized nice Corollary
Every finite family with the Erdős–Hajnal property is viral Theorem

Dependency tree · two levels

14 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

Erdős-Hajnal beyond the five-vertex path (standard reference, not scraped)