CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-28‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Strong Perfect Graph Theorem, ‡ Substituting perfect graphs preserves perfection and ‡ Weak Perfect Graph Theorem. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Every bull-free graph has a clique or stable set of size at least $∣ V (G) ∣^{1 / 4}$

Statement

Every bull-free finite graph $G$ contains a clique or a stable set of size at least $∣ V (G) ∣^{1 / 4}$ . Equivalently, the hereditary class of bull-free graphs has Erdős-Hajnal constant $1 / 4$ .

Facts & Assumptions

Given: A bull-free finite graph $G$ .

[L1]

Every bull-free graph is two-narrow (Every bull-free graph is 2-narrow).

[L2]

An $α$ -narrow graph has a clique or stable set of size at least $∣ V (G) ∣^{1 / (2 α)}$ (An $α$ -narrow graph has a clique or stable set of size at least $∣ V (G) ∣^{1 / (2 α)}$ ).

[F1]

The Erdős-Hajnal property is exactly the existence of a positive power lower bound for the homogeneous number (The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class, Real powers for positive bases, with the zero-base positive-exponent convention).

Proof

technique · direct

1.1

The bull-free theorem [L1] first gives that $G$ is two-narrow. Applying [L2] with $α = 2$ then yields a clique or stable set of size at least $∣ V (G) ∣^{1 / 4}$ .

L1L2algebra

2.1

This is exactly the graph-level form of an Erdős-Hajnal constant $1 / 4$ for the class of bull-free graphs, by [F1].

step 1.1F1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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, Theorem 2.3 (standard reference, not scraped)
Maria Chudnovsky and Shmuel Safra, The Erdős-Hajnal conjecture for bull-free graphs, Theorem 1.2 (standard reference, not scraped)
Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path (standard reference, not scraped)

Every bull-free graph has a clique or stable set of size at least ∣V(G)∣1/4∣V(G)∣1/4