CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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 four-vertex path has the Erdős-Hajnal property

Statement

The graph $P_{4}$ has the Erdős-Hajnal property.

Facts & Assumptions

Given: The four-vertex path $P_{4}$ .

[L1]

Every finite $P_{4}$ -free graph $G$ contains a clique or a stable set of size at least $\sqrt{∣ V (G) ∣}$ (Every $P_{4}$ -free graph has a clique or stable set of size at least the square root of its order).

[L2]

A graph $H$ has the Erdős-Hajnal property when the hereditary class of $H$ -free graphs has some positive Erdős-Hajnal constant (The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class).

Proof

technique · direct

1.1

By [L1], every nonempty $P_{4}$ -free graph $G$ satisfies $hom (G) \geq \sqrt{∣ V (G) ∣} = ∣ V (G) ∣^{1 / 2}$ , so the class of $P_{4}$ -free graphs has the positive exponent $1 / 2$ .

L1algebra

2.1

By [L2], the existence of that positive exponent is exactly the statement that $P_{4}$ has the Erdős-Hajnal property.

step 1.1L2∎

Depends on

Used by

Every graph on at most four vertices has the Erdős-Hajnal property Theorem

Dependency tree · two levels

16 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

Maria Chudnovsky, The Erdős-Hajnal Conjecture — A Survey, Section 2 (standard reference, not scraped)
Tung H. Nguyen, Notes on Recent Work on the Erdős-Hajnal Conjecture, Section 1 (standard reference, not scraped)