LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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.

If $G$ has fewer than $(δ n)^{h}$ induced copies of $H$ and $∣ W ∣ \geq λ n$ , then $G [W]$ has fewer than $((δ / λ) ∣ W ∣)^{h}$

Statement

Let $H$ have $h$ vertices. If $G$ has $n$ vertices, ${ind}_{H} (G) < (δ n)^{h}$ , and $W \subseteq V (G)$ satisfies $∣ W ∣ \geq λ n > 0$ , then

${ind}_{H} (G [W]) < ((δ / λ) ∣ W ∣)^{h} .$

Facts & Assumptions

Given: A graph $H$ with $h$ vertices, a graph $G$ on $n$ vertices, reals $δ > 0$ and $λ > 0$ , and a subset $W \subseteq V (G)$ with $∣ W ∣ \geq λ n$ and ${ind}_{H} (G) < (δ n)^{h}$ .

[L1]

An induced embedding of $H$ into $G [W]$ is, by definition, an induced embedding of $H$ into $G$ whose image lies in $W$ (Induced embeddings and induced copies of a graph, Subgraphs, induced subgraphs and spanning subgraphs).

[L2]

The induced-copy number counts induced embeddings (The induced-embedding count ${ind}_{H} (G)$ ).

Proof

technique · direct

1.1L1

By [L1], every induced embedding counted by ${ind}_{H} (G [W])$ is also counted by ${ind}_{H} (G)$ .

2.1step 1.1L2

Therefore ${ind}_{H} (G [W]) \leq {ind}_{H} (G) < (δ n)^{h}$ by [L2].

3.1step 2.1algebra∎

Since $∣ W ∣ \geq λ n$ , one has $n \leq ∣ W ∣ / λ$ . Substituting this into step 2.1 yields ${ind}_{H} (G [W]) < ((δ / λ) ∣ W ∣)^{h}$ .

Depends on

Used by

A linearly large induced subgraph of a graph with few induced copies again has a linearly large restricted set Corollary

Dependency tree · two levels

12 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

Y. Huang, Q. Ju, and X. Zhou, Erdős-Hajnal beyond the five-vertex path, sec. 1.1 (standard reference, not scraped)

If GG has fewer than (δn)h(δn)h induced copies of HH and ∣W∣≥λn∣W∣≥λn, then G[W]G[W] has fewer than ((δ/λ)∣W∣)h((δ/λ)∣W∣)h