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.

Sparsity is preserved when the parameter grows, and every nonempty set of at most two vertices is $0$ -restricted

Statement

Let $G$ be a finite simple graph and let $0 \leq c \leq c^{'}$ .

If a nonempty set $X \subseteq V (G)$ is $c$ -sparse, then it is $c^{'}$ -sparse.
If $X$ is nonempty and $∣ X ∣ \leq 2$ , then $X$ is $0$ -restricted.

Facts & Assumptions

Given: A finite simple graph $G$ , reals $0 \leq c \leq c^{'}$ , and a nonempty set $X \subseteq V (G)$ .

[L1]

A nonempty set $Y$ is $c$ -sparse when every vertex of $Y$ has at most $c ∣ Y ∣$ neighbours in $Y$ ; it is $c$ -dense when every vertex has at most $c ∣ Y ∣$ non-neighbours in $Y$ other than itself; and it is $c$ -restricted when it is $c$ -sparse or $c$ -dense ( $c$ -sparse, $c$ -dense and $c$ -restricted vertex sets).

Proof

technique · direct

1.1L1algebra

If $X$ is $c$ -sparse, then every vertex of $X$ has at most $c ∣ X ∣ \leq c^{'} ∣ X ∣$ neighbours in $X$ , so [L1] makes $X$ $c^{'}$ -sparse.

1.2L1

If $∣ X ∣ = 1$ , then its only vertex has no neighbour and no non-neighbour inside $X ∖ {x}$ , so [L1] makes $X$ both $0$ -sparse and $0$ -dense.

2.1L1∎

If $∣ X ∣ = 2$ , then either the two vertices are adjacent or they are not. In the first case $X$ is $0$ -dense, and in the second it is $0$ -sparse. So [L1] makes every two-element set $0$ -restricted.

Depends on

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

Used by

Nikiforov: for every H and every ε∈(0,1/2) there is δ>0 such that every graph G with ind_H(G)<(δ|V(G)|)^|V(H)| has an ε-restricted vertex set of size at least δ|V(G)| Theorem

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

M. Chudnovsky, A. Scott, P. Seymour, and S. Spirkl, Strengthening Rödl's theorem, sec. 1 (standard reference, not scraped)

Sparsity is preserved when the parameter grows, and every nonempty set of at most two vertices is 00-restricted