Ramsey repetition property at fixed k

PDF p. 2: Section 2, paragraph before Lemma 2.1

Fix a positive integer $k$. In this treatment a graph is a finite simple graph with a nonempty vertex set. A graph $H$ has the Ramsey repetition property at $k$ if, for every positive integer $q$ and every graph $A$, $qA\in {\mathcal{R}}_{\mathrm{i}\mathrm{n}\mathrm{d}}^k(H)$ if and only if $A\in {\mathcal{R}}_{\mathrm{i}\mathrm{n}\mathrm{d}}^k(H)$.

Here $qA$ is the disjoint union of $q$ copies of $A$: its vertices are pairs $(i,v)$ with $1\le i\le q$ and $v\in V(A)$, and two vertices are adjacent exactly when their first coordinates agree and their second coordinates are adjacent in $A$. For a graph $G$, membership $G\in {\mathcal{R}}_{\mathrm{i}\mathrm{n}\mathrm{d}}^k(H)$ means that every function $c:V(G)\to \{1,\dots ,k\}$ has a monochromatic induced copy of $H$. An induced copy is a set of vertices whose induced graph is isomorphic to $H$, preserving both adjacency and nonadjacency; monochromatic means that $c$ is constant on that set. The colouring may leave colours unused and need not assign different colours to adjacent vertices.

The graph conventions give definite vertex sets, adjacency and induced copies. The disjoint-multiple definition gives the displayed tagged graph $qA$, which is finite, simple and nonempty for the stated $A$ and $q$. The colouring definition specifies the unrestricted functions and the monochromatic predicate. The induced vertex-Ramsey class definition therefore gives a definite membership predicate for both $A$ and $qA$. Quantifying the equivalence over positive integers $q$ and graphs $A$ consequently defines a definite property of $H$ at the fixed $k$; it does not assert that every graph has the property.

Graph and colouring conventions

The disjoint multiple qA

Induced vertex-Ramsey class

Unrestricted finite colourings and used palettes