ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-09-05

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 four-tooth comb with a special vertex realizes the trigger configuration for property (*)

Example

Let $G$ have vertices

v, a_{1}, a_{2}, a_{3}, a_{4}, b_{i, j} (i, j \in [4])

with edges exactly $a_{i} b_{i, j}$ and $v b_{i, j}$ for $i, j \in [4]$ . Put

B_{i} : = {b_{i, 1}, b_{i, 2}, b_{i, 3}, b_{i, 4}}

for each $i$ .

Facts & Assumptions

Given: The graph and the blocks displayed in the Example.

[L1]

The item Combs in a graph characterizes an $(ℓ, w)$ -comb by the blockade conditions and the adjacency pattern of the teeth.

[L2]

If a finite family $F$ has property $(*)$ and $G$ is $\overline{F}$ -free, then an $(ℓ, w)$ -comb with $ℓ, w \geq 4$ together with a vertex outside the comb that is complete to the blocks and anticomplete to the teeth is the antecedent of the three-outcome implication in Property (*) for a finite graph family.

Verification

technique · direct

1.1

The four sets $B_{1}, B_{2}, B_{3}, B_{4}$ are pairwise disjoint and each has $4$ vertices, so $(B_{1}, B_{2}, B_{3}, B_{4})$ is a $(4, 4)$ -blockade.

givenL1

2.1

Each tooth $a_{i}$ is adjacent to every vertex of $B_{i}$ and to no vertex of $B_{j}$ for $j \neq i$ , and the teeth $a_{1}, a_{2}, a_{3}, a_{4}$ are distinct and lie outside the blocks. Hence $((a_{i}, B_{i}) : i \in [4])$ is a $(4, 4)$ -comb by [L1].

step 1.1L1

3.1

The vertex $v$ lies outside the comb, is adjacent to every vertex in $⋃_{i = 1}^{4} B_{i}$ , and is nonadjacent to every $a_{i}$ . Since this comb has $ℓ = w = 4$ , the pair consisting of the comb and $v$ realizes the geometric trigger configuration occurring in [L2]. No assertion that an unspecified family has property $(*)$ is being made.

step 2.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

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

Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Section 1.4 (standard reference, not scraped)