ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck 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.

For positive $a, b$ , $hom (K_{a, b}) = \max {2, a, b}$

Example

If $a, b$ are positive integers, then $hom (K_{a, b}) = \max {2, a, b} .$

Facts & Assumptions

Given: Positive integers $a, b$ and a bipartition $A \cup B$ of $K_{a, b}$ with $∣ A ∣ = a$ and $∣ B ∣ = b$ .

[L1]

The homogeneous number is the maximum of the clique and stable-set numbers (Homogeneous vertex sets and the homogeneous number $hom (G) = \max {ω (G), α (G)}$ ).

[L2]

In $K_{a, b}$ , the parts $A, B$ are disjoint, all cross-pairs are edges, and there are no edges inside either part (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

Verification

technique · direct

1.1givenL2

A clique uses at most one vertex from each part, while positivity of $a, b$ supplies a cross-edge, so $ω (K_{a, b}) = 2$ .

1.2L2

A stable set lies wholly in one part, and either whole part is stable, so $α (K_{a, b}) = \max {a, b}$ .

2.1step 1.1step 1.2L1∎

Taking the maximum in [L1] gives the formula.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

M. Chudnovsky, The Erdos-Hajnal Conjecture: A Survey, sec. 1 (standard reference, not scraped)

For positive a,ba,b, hom⁡(Ka,b)=max⁡{2,a,b}hom(Ka,b​)=max{2,a,b}