ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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 two sides of a balanced complete bipartite graph are large restricted sets

Example

In the balanced complete bipartite graph $K_{m, m}$ with $m \geq 1$ , each side is $0$ -sparse and therefore restricted; a set taking linearly many vertices from both sides is not $ϵ$ -restricted when $0 \leq ϵ < 1 / 2$ .

Facts & Assumptions

Given: The complete bipartite graph $K_{m, m}$ with $m \geq 1$ and bipartition $A ⊔ B$ , a real $0 \leq ϵ < 1 / 2$ , and a set $X$ meeting each side in exactly $a$ vertices.

[L1]

Each side of a complete bipartite graph is stable and therefore $0$ -sparse (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices, The $0$ -sparse sets are exactly the stable sets and the $0$ -dense sets exactly the cliques).

[L2]

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

Verification

technique · direct

1.1

By [L1], each of $A$ and $B$ is $0$ -sparse, so each is a restricted set of size $m$ .

1.2

If $X$ takes $a$ vertices from each side, then every vertex of $X$ has exactly $a$ neighbours and $a - 1$ non-neighbours inside $X$ , while $∣ X ∣ = 2 a$ .

given

2.1

For $0 \leq ϵ < 1 / 2$ and large $a$ , neither inequality $a \leq ϵ (2 a)$ nor $a - 1 \leq ϵ (2 a)$ can hold. Hence such balanced mixed sets are not $ϵ$ -restricted.

step 1.2L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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.