TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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 forest-free graph has a linear anticomplete pair or a linear-degree vertex

Statement

For every forest $H$ , there exists a real constant $ϵ_{H} > 0$ such that every finite $H$ -free graph $G$ with $∣ V (G) ∣ \geq 2$ satisfies at least one of the following:

some vertex of $G$ has degree at least $ϵ_{H} ∣ V (G) ∣$ ;
there exist disjoint sets $A, B \subseteq V (G)$ with $∣ A ∣ \geq ϵ_{H} ∣ V (G) ∣, ∣ B ∣ \geq ϵ_{H} ∣ V (G) ∣,$ and $A$ anticomplete to $B$ .

Facts & Assumptions

Given: A forest $H$ and a finite $H$ -free graph $G$ with $∣ V (G) ∣ \geq 2$ .

[F1]

For every forest $H$ , the source theorem supplies a real constant $ϵ_{H} > 0$ such that every finite $H$ -free graph on at least two vertices has a vertex of degree at least $ϵ_{H} ∣ V (G) ∣$ or has two disjoint anticomplete sets $A, B$ with $∣ A ∣ \geq ϵ_{H} ∣ V (G) ∣, ∣ B ∣ \geq ϵ_{H} ∣ V (G) ∣ .$

Proof

technique · direct

1.1

By [F1], choose the constant $ϵ_{H} > 0$ attached to the forest $H$ . For the given graph $G$ , the same source theorem gives at least one of the two alternatives in the Statement.

F1choose

2.1

Therefore $ϵ_{H}$ has the required property for all finite $H$ -free graphs with at least two vertices.

step 1.1∎

Depends on

Used by

A forest complement and its star-expansion have the Erdős-Hajnal property Theorem

Dependency tree · two levels

10 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, Alex Scott, Paul Seymour, and Sophie Spirkl, Pure pairs. I. Trees and linear anticomplete pairs, statement 1.4 (standard reference, not scraped)