LemmaStatement: AI-adaptedProof: AI-generated

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.

Fewer colours for the base graph

PDF p. 3: Section 3, paragraph beginning First, final sentence

Statement

Let $A$ and $F$ be nonempty finite simple graphs, and let $1 \leq i \leq k$ be integers. If $A \in R_{i n d}^{k} (F)$ , then $A \in R_{i n d}^{i} (F)$ . In particular, the base graph $A$ supplied by Existence of the base graph A belongs to $R_{i n d}^{i} (F)$ for every $i \in {2, \dots, k}$ .

Here $R_{i n d}^{j} (F)$ consists of the nonempty finite simple graphs whose every vertex colouring by a function into $[j] = {1, \dots, j}$ has a monochromatic induced copy of $F$ : its image vertices all have the same colour and their adjacency and nonadjacency agree with those of $F$ . A colouring need not use every available colour. Graph and colouring conventions Induced vertex-Ramsey class Unrestricted finite colourings and used palettes

Facts & Assumptions

[F1]

For positive integers $j$ , membership in $R_{i n d}^{j} (F)$ means that every function $V (A) \to [j]$ has a monochromatic induced copy of $F$ . Induced vertex-Ramsey class

[F2]

A finite colouring is a function into its palette, with no requirement to use every colour. Injective relabelling of its used palette preserves and reflects monochromatic vertex subsets. Unrestricted finite colourings and used palettes Transport, restriction and relabelling of finite colourings

[F3]

For natural numbers $i \leq k$ , $i \subseteq k$ ; with $[j] = σ (j) ∖ {0}$ , this gives $[i] \subseteq [k]$ . On ℕ the order is membership: m < n ⟺ m ∈ n Unrestricted finite colourings and used palettes

[F4]

Under its stated hypotheses, the base-graph proposition supplies a nonempty finite simple graph $A$ belonging to $R_{i n d}^{k} (F)$ . Existence of the base graph A

Proof

Given: Nonempty finite simple graphs $A, F$ , positive integers $i \leq k$ , and $A \in R_{i n d}^{k} (F)$ .

Goal: Show $A \in R_{i n d}^{i} (F)$ and deduce the stated base-graph instance.

1.1givenF2F3

Let $c : V (A) \to [i]$ be any $i$ -colouring. By F3, inclusion maps $[i]$ injectively into $[k]$ . Restricting this inclusion to the used palette of $c$ gives an injective relabelling, so the resulting map $d : V (A) \to [k]$ has exactly the same monochromatic vertex subsets as $c$ , by F2.

2.1givenF1step 1.1

By F1 and the given membership, $d$ has a monochromatic induced copy of $F$ in $A$ . Its vertex set is also monochromatic for $c$ by step 1.1; changing colour labels does not change the graph or the induced copy. Since $c$ was arbitrary, F1 gives $A \in R_{i n d}^{i} (F)$ .

3.1F4step 2.1∎

Under the hypotheses of the base-graph proposition, F4 supplies its required $k$ -colour membership. Applying step 2.1 to each integer $i$ with $2 \leq i \leq k$ gives the article's stated instance; if there is no such $i$ , that instance is vacuous.

Depends on

Used by · 1

A monochromatic F extends through all-colour attachments

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

Sources

Maria Axenovich and Patrick Schmitteckert, Non-isomorphic graphs have distinct vertex-Ramsey classes, arXiv:2610.03704v1 · PDF p. 3, Section 3, paragraph beginning First, final sentence