LemmaStatement: AI-adaptedProof: AI-generated

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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.

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Copies on hyperedges are induced

PDF p. 2: Construction 1, last two sentences

Statement

Let $H$ be a finite simple graph with at least one vertex, let $q \geq 3$ and $k \geq 1$ be integers, and fix the choices defining $G = G (H, q, k)$ in Construction 1: graph G(H,q,k). Write $H = (V, E)$ for its underlying hypergraph, $f_{e} : V (H) \to e$ for the chosen bijection, and $J_{e} = (e, E_{e})$ for the transported piece on each $e \in E$ . Then the induced subgraph $G [e]$ equals $J_{e}$ . Consequently $f_{e}$ is an induced embedding of $H$ in $G$ , and $G [e]$ is an induced copy of $H$ .

Facts & Assumptions

[F1]

By Construction 1: graph G(H,q,k), $H$ is finite of Berge girth at least $q$ , each $f_{e}$ is a bijection, $E_{e} = {{f_{e} (x), f_{e} (y)} : {x, y} \in E (H)}$ , and $G = (V, ⋃_{e \in E} E_{e})$ . Thus $G$ is the Graph realisations with possibly different hyperedge pieces of the pieces $J_{e}$ .

[F2]

By Berge girth at least three forces linearity, distinct hyperedges of $H$ intersect in at most one vertex. This is linearity in Berge cycles, incidence graphs and girth.

[F3]

By Linear realisations preserve induced pieces, a finite linear hypergraph realisation induces each of its pieces; when the pieces are transported from a graph by bijections, those bijections are induced embeddings. Here $G [e]$ is the induced subgraph on $e$ , as in Subgraphs, induced subgraphs and spanning subgraphs; the terms bijection and induced embedding and copy are as in Injection, surjection, bijection and Induced embeddings and induced copies of a graph.

Proof

Given: The graph $H$ , integers $q, k$ , and fixed construction choices in the statement.

Goal: For every $e \in E$ , prove $G [e] = J_{e}$ and that $f_{e}$ exhibits an induced copy of $H$ .

1.1givenF1F2

The construction gives a finite hypergraph of Berge girth at least $q \geq 3$ . Its distinct hyperedges intersect in at most one vertex, so it is linear.

2.1F1F3step 1.1∎

The graph $G$ is the hyperedge-indexed realisation of the pieces transported from $H$ by the bijections $f_{e}$ . Applying the linear-realisation lemma to this finite linear hypergraph gives $G [e] = J_{e}$ for every $e$ , and shows that each $f_{e}$ is an induced embedding whose image induces a copy of $H$ . This is the desired conclusion.

Depends on

Used by · 1

Construction 1 gives an induced vertex-Ramsey graph

Dependency tree · 2 levels

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Sources

Maria Axenovich and Patrick Schmitteckert, Non-isomorphic graphs have distinct vertex-Ramsey classes, arXiv:2610.03704v1 · PDF p. 2, Construction 1, last two sentences