DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-26

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

The $(t, k)$ -homogeneous property

Definition

Let $t$ and $k$ be natural numbers. A finite graph $G$ has the $(t, k)$ -homogeneous property if every $t$ -element subset $X \subseteq V (G)$ contains a homogeneous subset $Y \subseteq X$ with $∣ Y ∣ = k$ (Homogeneous vertex sets and the homogeneous number $hom (G) = \max {ω (G), α (G)}$ ).

A class $C$ of finite graphs has the $(t, k)$ -homogeneous property if every graph in $C$ has it.

Remarks

When $∣ V (G) ∣ < t$ , the condition on $t$ -element subsets is vacuous.
The property is designed to be used on exact $t$ -vertex induced subgraphs: later proofs first build such a subgraph and then extract the homogeneous $k$ -set from it.

Depends on

Homogeneous vertex sets and the homogeneous number $\operatorname{hom}(G)=\max\{\omega(G),\alpha(G)\}$

Used by

Dependency tree · two levels

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Sources

M. Bucić, J. Fox, and H. T. Pham, Equivalence between Erdős-Hajnal and polynomial Rödl and Nikiforov conjectures, Lemma 13 (standard reference, not scraped)

The (t,k)(t,k)-homogeneous property