Nonemptiness attributed to Folkman

By Graph and colouring conventions, a graph here has a nonempty finite vertex set and an edge set consisting of two-element subsets. A vertex $k$-colouring is any function into $\{1,\dots ,k\}$. An induced copy of $H$ in a graph $G$ preserves both adjacency and nonadjacency, and is monochromatic when its vertices all have the same colour.

By Induced vertex-Ramsey class, $G\in {\mathcal{R}}_{\mathrm{i}\mathrm{n}\mathrm{d}}^k(H)$ exactly when $G$ is a nonempty finite simple graph and every vertex $k$-colouring of $G$ has a monochromatic induced copy of $H$.

For a finite graph $H$ of order at least two, High-girth hypergraphs for Construction 1 supplies a finite $\mathrm{\mid }V(H)\mathrm{\mid }$-uniform hypergraph of Berge girth at least three with no weak $k$-colouring. Here uniformity means each hyperedge has $\mathrm{\mid }V(H)\mathrm{\mid }$ vertices; no weak $k$-colouring means every assignment of the $k$ colours to its vertices has a monochromatic hyperedge. By Construction 1: graph G(H,q,k), this hypergraph and bijections from $V(H)$ to its hyperedges give choices for the graph $G(H,3,k)$. The hypergraph terms, including Berge cycles and Berge girth, have the meanings in Hypergraph terminology used in Construction 1, and bijections have the meaning in Injection, surjection, bijection.

By Construction 1 gives an induced vertex-Ramsey graph, every graph obtained from those construction choices belongs to ${\mathcal{R}}_{\mathrm{i}\mathrm{n}\mathrm{d}}^k(H)$.