CounterexampleConstruction: Literature-sourcedVerification: 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.

A star has tiny self-density, yet no restricted subset containing its centre has more than two vertices

Statement refuted

Every weakly sparse set is sparse.

Facts & Assumptions

Given: A real $ϵ < 1 / 3$ , an integer $m \geq 2$ , the star $K_{1, m}$ with centre $v$ , and its full vertex set $X$ .

[L1]

A set is $ϵ$ -sparse or $ϵ$ -dense according to the degree and non-neighbour bounds of $c$ -sparse, $c$ -dense and $c$ -restricted vertex sets.

[L2]

The self-density is computed from the ordered internal edge count (Edge counts and densities between nonempty vertex sets).

Counterexample

technique · constructive

1.1

The set $X$ has $m + 1$ vertices and exactly $m$ edges, so $d_{G} (X, X) = 2 m / (m + 1)^{2}$ , which tends to $0$ as $m \to \infty$ .

L2givenconstruct

2.1

Let $Y \subseteq X$ contain the centre $v$ and at least two leaves. Then $v$ has $∣ Y ∣ - 1$ neighbours in $Y$ , so the sparse inequality in [L1] fails when $ϵ < 1 / 3$ .

step 1.1L1algebra

3.1

Each leaf of $Y$ has at least $∣ Y ∣ - 2$ non-neighbours in $Y$ , so the dense inequality in [L1] also fails when $ϵ < 1 / 3$ . Thus no such $Y$ is $ϵ$ -restricted.

step 2.1L1algebra

4.1

Hence a set can have arbitrarily small self-density without being sparse or dense in the maximum-degree sense.

step 1.1step 3.1discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

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