LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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.

Iterated sparse restriction reaches the target sparsity threshold

Statement

Let $c \in (0, 1)$ , $b_{1} > 1$ , $b_{2}, b_{3} > 0$ , and assume $b_{1} b_{2} \geq b_{2} + b_{3} .$ Suppose that $x \in (0, c)$ and that a graph $G$ satisfies:

$G$ has a $c$ -sparse induced subgraph with at least $c^{b_{2}} ∣ V (G) ∣$ vertices; and
for every $λ \in [x, c]$ and every $λ$ -sparse induced subgraph $F$ of $G$ with $∣ V (F) ∣ \geq λ^{b_{2}} ∣ V (G) ∣$ , there is a $λ^{b_{1}}$ -sparse induced subgraph of $F$ with at least $λ^{b_{3}} ∣ V (F) ∣$ vertices.

Then $G$ contains an $x$ -sparse induced subgraph with at least $x^{b_{1} b_{2}} ∣ V (G) ∣$ vertices.

Facts & Assumptions

Given: The parameters and hypotheses in the statement.

[L1]

A $λ$ -sparse vertex set is nonempty, and every vertex has degree at most $λ$ times the size of that set inside the induced subgraph ( $c$ -sparse, $c$ -dense and $c$ -restricted vertex sets).

Proof

technique · direct

1.1

Hypothesis 1 supplies a nonempty $c$ -sparse vertex set, because $c$ -sparse sets are nonempty by [L1]. Hence $∣ V (G) ∣ > 0$ .

givenL1

1.2

For each nonempty induced subgraph $E$ of $G$ , let $μ (E)$ be its maximum degree divided by $∣ V (E) ∣$ ; by [L1], the graph $E$ is $λ$ -sparse exactly when $μ (E) \leq λ$ . Define $λ (E) : = \max (x^{b_{1}}, μ (E))$ . Hypothesis 1 gives a $c$ -sparse induced subgraph $E_{0}$ with at least $c^{b_{2}} ∣ V (G) ∣$ vertices, so $λ (E_{0}) \leq c$ and $∣ V (E_{0}) ∣ \geq λ (E_{0})^{b_{2}} ∣ V (G) ∣$ . Because $G$ has only finitely many induced subgraphs, the set of values $λ (E)$ with $λ (E) \leq c$ and $∣ V (E) ∣ \geq λ (E)^{b_{2}} ∣ V (G) ∣$ has a minimum. Choose an induced subgraph $F$ for which that minimum is attained, and write $λ : = λ (F)$ .

givenchooseL1algebra

2.1

Suppose that $λ \geq x$ . Then hypothesis 2 applies to $F$ and produces a $λ^{b_{1}}$ -sparse induced subgraph $F^{'} \subseteq F$ with at least $λ^{b_{3}} ∣ V (F) ∣ \geq λ^{b_{2} + b_{3}} ∣ V (G) ∣ \geq λ^{b_{1} b_{2}} ∣ V (G) ∣$ vertices, where the last inequality uses $b_{1} b_{2} \geq b_{2} + b_{3}$ . Since $F^{'}$ is $λ^{b_{1}}$ -sparse, we have $μ (F^{'}) \leq λ^{b_{1}}$ and therefore $λ (F^{'}) = \max (x^{b_{1}}, μ (F^{'})) \leq λ^{b_{1}} < λ$ . Also $∣ V (F^{'}) ∣ \geq λ (F^{'})^{b_{2}} ∣ V (G) ∣$ . This contradicts the minimal choice of $λ$ . Therefore $λ < x$ .

step 1.2givenalgebraassume-contradischarge-contradiction

3.1

Since $λ = λ (F)$ , step 1.2 gives $x^{b_{1}} \leq λ$ , and step 2.1 gives $λ < x$ . Because $F$ is $λ$ -sparse, it is also $x$ -sparse. Moreover $∣ V (F) ∣ \geq λ^{b_{2}} ∣ V (G) ∣ \geq (x^{b_{1}})^{b_{2}} ∣ V (G) ∣ = x^{b_{1} b_{2}} ∣ V (G) ∣$ .

step 1.2step 2.1algebra

4.1

The induced subgraph $F$ from step 3.1 is the required $x$ -sparse induced subgraph.

step 3.1∎

Depends on

$c$-sparse, $c$-dense and $c$-restricted vertex sets

Used by

A numeric run of the multiplicative iteration in Lemma 2.9 Example

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

Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Lemma 2.9 (standard reference, not scraped)
Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density. VI. Bounded VC-dimension, Lemma 3.2 (standard reference, not scraped)