LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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 vertex mixed on a connected set has opposite adjacency on some edge of that set

Statement

If $S$ induces a connected graph and $u \notin S$ is mixed on $S$ , then some edge $a b$ of $G [S]$ has exactly one endpoint adjacent to $u$ .

Facts & Assumptions

Given: A connected set $S$ and a vertex $u$ mixed on it.

[F1]

Mixedness supplies a neighbour and a nonneighbour of $u$ in $S$ (Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs).

[F2]

Connected vertices are joined by a path in the induced graph (Connected graphs and connected components defined by the existence of vertex paths).

Proof

technique · direct

1.1

Choose $a, b \in S$ with $u a$ an edge and $u b$ a nonedge by [F1], and choose an $a$ -- $b$ path in $G [S]$ by [F2].

F1F2choose

2.1

Along this finite path, the adjacency indicator to $u$ begins at $1$ and ends at $0$ , so it first changes across one consecutive pair. That pair is an edge of $G [S]$ with opposite adjacencies to $u$ .

step 1.1algebra

3.1

This is the required mixed edge.

step 2.1∎

Depends on

Used by

In a special-vertex comb of a co-E-free graph, vertices in other comb blocks remain pure to every H₅-overlap quotient block Lemma

Dependency tree · two levels

8 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

Huang, Ju, and Zhou, Erdős-Hajnal beyond the five-vertex path, proof of Claim 6.4.3 (standard reference, not scraped)