CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

The dense alternative in Rödl's theorem cannot be dropped

Statement refuted

The dense alternative in Rödl's theorem is unnecessary.

Facts & Assumptions

Given: A real $0 \leq ϵ < 1 / 2$ and the complete graph $K_{n}$ with $n \geq 2$ .

[L1]

A set is $ϵ$ -sparse when each of its vertices has at most $ϵ ∣ X ∣$ neighbours inside it ( $c$ -sparse, $c$ -dense and $c$ -restricted vertex sets).

[L2]

A graph is $P_{3}$ -free when it has no induced three-vertex path ( $H$ -free and $F$ -free graphs under the induced-subgraph convention, Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

Counterexample

technique · constructive

1.1

Every nonempty subset $X$ of $K_{n}$ with $∣ X ∣ \geq 2$ has each vertex adjacent to all $∣ X ∣ - 1$ other vertices of $X$ .

givenconstruct

2.1

If such an $X$ were $ϵ$ -sparse, then [L1] would force $(1 - ϵ) ∣ X ∣ \leq 1$ . Hence every $ϵ$ -sparse subset of $K_{n}$ has size at most $1 / (1 - ϵ)$ , a bound independent of $n$ .

step 1.1L1algebra

3.1

The graph $K_{n}$ is $P_{3}$ -free, since every three vertices induce a triangle rather than a path. For any proposed positive linear constant $δ$ , choosing $n > 1 / (δ (1 - ϵ))$ makes every $ϵ$ -sparse set smaller than $δ n$ by step 2.1. Thus a linear restricted set in this $P_{3}$ -free family must use the dense alternative, which cannot be discarded.

step 2.1L2discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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.