TheoremStatement: Literature-sourcedProof: AI-generatedverified 2026-09-24 (gpt-6-sol)

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For every forest $H$ , graphs excluding $H$ and $\overline{H}$ have a linear pure pair

Statement

For every forest $H$ , there exists a real constant $ϵ_{H} > 0$ such that every finite graph $G$ with no induced $H$ and no induced $\overline{H}$ and with $∣ V (G) ∣ \geq 2$ contains disjoint sets $A, B \subseteq V (G)$ satisfying

$∣ A ∣ \geq ϵ_{H} ∣ V (G) ∣, ∣ B ∣ \geq ϵ_{H} ∣ V (G) ∣,$

and such that $(A, B)$ is a pure pair. Equivalently, the hereditary class of graphs forbidding $H$ and $\overline{H}$ has the strong Erdős-Hajnal property.

Facts & Assumptions

Given: A forest $H$ and a finite graph excluding both $H$ and $\overline{H}$ .

[L1]

There is $e_{H} > 0$ such that every $H$ -free graph on at least two vertices has an anticomplete pair with both sides at least $e_{H}$ times its order, or a vertex of degree at least $e_{H}$ times its order (Every forest-free graph has a linear anticomplete pair or a linear-degree vertex).

[L2]

For $c \in (0, 1 / 2)$ , every nonempty $H$ -free graph has a $c$ -restricted set of size at least $δ ∣ G ∣$ for some $δ > 0$ depending on $H, c$ (Rödl: for every $H$ and every $ϵ \in (0, \frac{1}{2})$ there is $δ > 0$ such that every nonempty $H$ -free graph has an $ϵ$ -restricted vertex set of size at least $δ ∣ V (G) ∣$ ).

[L3]

The forbidden induced-subgraph class is hereditary (Every class defined by forbidden induced subgraphs is hereditary).

Proof

technique · apply Rödl's restricted-set theorem and the forest dichotomy

1.1

Choose $0 < e \leq e_{H}$ with $e < 1 / 2$ and apply [L2] with $c = e / 2$ , obtaining $δ > 0$ . Put $ϵ_{H} = \min {e δ, δ} > 0$ .

L1L2choose

2.1

Let $G$ exclude $H, \overline{H}$ and have $n \geq 2$ vertices. By [L2] it has $X$ with $∣ X ∣ \geq δ n$ which is either $(e / 2)$ -sparse or $(e / 2)$ -dense. If $∣ X ∣ = 1$ , then $δ n \leq 1$ ; any two distinct vertices of $G$ are a complete or anticomplete pair of singletons, both of size at least $ϵ_{H} n$ .

step 1.1L2algebra

3.1

Suppose $∣ X ∣ \geq 2$ and $X$ is $(e / 2)$ -sparse. The induced graph $G [X]$ is $H$ -free and has maximum degree at most $(e / 2) ∣ X ∣ < e_{H} ∣ X ∣$ . Therefore [L1] gives disjoint anticomplete $A, B \subseteq X$ with $∣ A ∣, ∣ B ∣ \geq e_{H} ∣ X ∣ \geq ϵ_{H} n$ .

step 1.1step 2.1L1

3.2

Suppose instead $∣ X ∣ \geq 2$ and $X$ is $(e / 2)$ -dense. The complement $\overline{G [X]}$ is $H$ -free because $G$ excludes $\overline{H}$ , and its maximum degree is at most $(e / 2) ∣ X ∣$ . Apply [L1] there to obtain an anticomplete pair of size at least $e_{H} ∣ X ∣$ on each side; it is a complete pair of size at least $ϵ_{H} n$ in $G$ .

step 1.1step 2.1L1

4.1

All cases give a pure pair of the required linear size. By [L3] and the definition of strong Erdős–Hajnal property, the hereditary class excluding $H, \overline{H}$ has that property.

step 2.1step 3.1step 3.2L3∎

Depends on

Used by

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Sources

Maria Chudnovsky, Alex Scott, Paul Seymour, and Sophie Spirkl, Pure pairs. I. Trees and linear anticomplete pairs, statement 1.2 (standard reference, not scraped)

For every forest HH, graphs excluding HH and H‾H have a linear pure pair