LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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Pure blockades with $P_{4}$ -free patterns contain complete or anticomplete subblockades of square-root length

Statement

Let $B = (B_{1}, \dots, B_{t})$ be a pure blockade whose pattern graph is $P_{4}$ -free. Then $B$ has a complete or anticomplete subblockade of length at least $\sqrt{t}$ and of width at least the width of $B$ .

Facts & Assumptions

Given: A pure blockade $B = (B_{1}, \dots, B_{t})$ with $P_{4}$ -free pattern graph $P (B)$ .

Proof

technique · direct

1.1given

By A $P_{4}$ -free graph on $q$ vertices has a homogeneous set of size at least $\sqrt{q}$ , the pattern graph $P (B)$ has a clique or stable set $I \subseteq [t]$ with $∣ I ∣ \geq \sqrt{t}$ .

2.1step 1.1given

If $I$ is a clique, then by the definition of the pattern graph every pair of blocks indexed by $I$ is complete, so $(B_{i} : i \in I)$ is a complete subblockade. If $I$ is a stable set, the same definition makes $(B_{i} : i \in I)$ anticomplete. In either case the width does not decrease when blocks are discarded.

3.1step 2.1∎

Therefore $B$ contains a complete or anticomplete subblockade of length at least $\sqrt{t}$ and of at least the original width.

Depends on

Used by

Complete or anticomplete blockade hypotheses force an ε-restricted induced subgraph Theorem

Dependency tree · two levels

6 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, Erdős-Hajnal for graphs with no 5-hole, §5 (standard reference, not scraped)

Pure blockades with P4P4​-free patterns contain complete or anticomplete subblockades of square-root length