LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26

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Small total induced-copy expectation forces many homogeneous $k$ -sets

Statement

Let $1 \leq k \leq t$ be integers, and let $F$ be a finite family of graphs, each with at least one vertex, such that every $F$ -free graph has the $(t, k)$ -homogeneous property. Let $G$ be a finite graph on $n \geq 2 t$ vertices. Choose $X$ uniformly from $[V (G)]^{2 t}$ and define

Y (X) : = \sum_{H \in F} {ind}_{H} (G [X]) .

If $E [Y] \leq t / 2$ , then $G$ has at least

\frac{1}{2} {(\frac{n}{2 t})}^{k}

homogeneous vertex sets of size $k$ .

Facts & Assumptions

Given: Positive integers $1 \leq k \leq t$ , a finite family $F$ of graphs, each with at least one vertex, a finite graph $G$ on $n \geq 2 t$ vertices, the uniform choice of $X \in [V (G)]^{2 t}$ , and the hypothesis $E [Y] \leq t / 2$ .

[L1]

A graph is $F$ -free exactly when it is $H$ -free for every $H \in F$ ( $H$ -free and $F$ -free graphs under the induced-subgraph convention).

[L2]

$Y (X)$ is a nonnegative real random variable on the uniform probability space on $[V (G)]^{2 t}$ , and its expectation is the average value over that finite outcome set (The uniform probability space on a nonempty finite set, Expectation of a real random variable on a finite probability space, The induced-embedding count ${ind}_{H} (G)$ ).

[L3]

If a nonnegative random variable has expectation at most $t / 2$ , then the probability that it is at least $t$ is at most $1 / 2$ (Markov's inequality on a finite probability space).

[L4]

If at least half of the $2 t$ -element subsets of $V (G)$ contain a $t$ -element induced subgraph in a class with the $(t, k)$ -homogeneous property, then $G$ has at least $\frac{1}{2} (n / (2 t))^{k}$ homogeneous $k$ -sets (Many good $2 t$ -vertex subsets force many homogeneous $k$ -sets).

Proof

technique · direct

1.1

Since $Y$ is nonnegative and $E [Y] \leq t / 2$ , [L3] gives $P (Y \geq t) \leq 1 / 2$ , so with probability at least $1 / 2$ one has $Y < t$ .

L2L3

2.1

Fix a set $X \in [V (G)]^{2 t}$ with $Y (X) < t$ . For each induced embedding counted by $Y (X)$ choose one vertex from its image; this is possible because every graph in $F$ has at least one vertex. Delete from $X$ every chosen vertex. Since fewer than $t$ embeddings were counted, fewer than $t$ vertices are deleted, so at least $t$ vertices remain.

step 1.1choose

3.1

Let $T$ be any $t$ -element subset of the remaining vertices. If some $H \in F$ had an induced embedding into $G [T]$ , then that same embedding would already have been counted in $Y (X)$ , so step 2.1 would have deleted a vertex from its image. Because the image lies in $T$ , this contradicts the choice of $T$ . Thus $G [T]$ is $F$ -free by [L1].

step 2.1L1choose

4.1

Steps 2.1 and 3.1 show that with probability at least $1 / 2$ , a uniformly random $2 t$ -element subset of $V (G)$ contains a $t$ -element induced subgraph that is $F$ -free. Applying [L4] to the class of $F$ -free graphs proves the claimed lower bound on homogeneous $k$ -sets.

step 1.1step 3.1L4∎

Depends on

Used by

Every finite family with the Erdős–Hajnal property is viral Theorem

Dependency tree · two levels

21 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. Bucić, J. Fox, and H. T. Pham, Equivalence between Erdős-Hajnal and polynomial Rödl and Nikiforov conjectures, Lemma 14 (standard reference, not scraped)
T. H. Nguyen, Notes on Recent Work on the Erdős–Hajnal Conjecture, Lemma 14 (standard reference, not scraped)

Small total induced-copy expectation forces many homogeneous kk-sets