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

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A wide coherent blockade contains a blockade-rainbow copy of a forest

Statement

Let $F$ be a forest on vertices $u_{1}, \dots, u_{m}$ , and let $B = (B_{1}, \dots, B_{m})$ be a blockade in a graph $G$ . Suppose that for every distinct $i, j \in [m]$ ,

$B_{i}$ is complete to $B_{j}$ when $u_{i} u_{j} \in E (F)$ ; and
$B_{i}$ is anticomplete to $B_{j}$ when $u_{i} u_{j} \notin E (F)$ .

Then $G$ contains a $B$ -rainbow induced copy of $F$ .

Facts & Assumptions

Given: A forest $F$ on vertices $u_{1}, \dots, u_{m}$ , a graph $G$ , and a blockade $B = (B_{1}, \dots, B_{m})$ satisfying the two displayed cross-block conditions.

[L1]

A $B$ -rainbow induced copy of $F$ means an induced copy lying in $V (B)$ and using at most one vertex from each block (A blockade-rainbow induced copy).

[F1]

Every block of a blockade is nonempty (Blockades, their length, their width, and their support).

Proof

technique · direct

1.1

By [F1], choose vertices $x_{i} \in B_{i}$ for every $i \in [m]$ . Let $X : = {x_{1}, \dots, x_{m}}$ . Because the blocks are pairwise disjoint, these $m$ vertices are distinct.

F1choosegiven

2.1

For distinct $i, j \in [m]$ , the hypothesis says that $x_{i}$ and $x_{j}$ are adjacent exactly when $u_{i}$ and $u_{j}$ are adjacent in $F$ . Hence the map $u_{i} \mapsto x_{i}$ is an adjacency-preserving and nonadjacency-preserving bijection from $V (F)$ to $X$ , so $G [X]$ is an induced copy of $F$ .

step 1.1given

3.1

The copy $G [X]$ lies in $V (B)$ and uses exactly one vertex from each block, so [L1] shows that it is $B$ -rainbow.

step 2.1L1∎

Depends on

Used by

A long blockade without a large pure pair contains a rainbow forest or its complement 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, Theorem 2.1 (standard reference, not scraped)
Maria Chudnovsky, Alex Scott, Paul Seymour, and Sophie Spirkl, Erdős-Hajnal for graphs with no 5-hole, Theorem 6.3 (standard reference, not scraped)