Theorem 1.1: both vertex-Ramsey distinctions

PDF p. 1: Abstract; Theorem 1.1; PDF p. 2: Section 3, ordinary part; PDF p. 3: Section 3, induced part; PDF p. 4: Section 3, conclusion

Let $H_1,H_2$ be non-isomorphic finite simple graphs, each with at least one vertex, and let $k$ be a positive integer. Then ${\mathcal{R}}^k(H_1)\mathrm{\ne }{\mathcal{R}}^k(H_2)$ and ${\mathcal{R}}_{\mathrm{i}\mathrm{n}\mathrm{d}}^k(H_1)\mathrm{\ne }{\mathcal{R}}_{\mathrm{i}\mathrm{n}\mathrm{d}}^k(H_2)$.

Graphs are isomorphic if a bijection between their vertex sets preserves and reflects adjacency. For a graph $H$, ${\mathcal{R}}^k(H)$ is the collection of nonempty finite simple graphs $G$ for which every function $V(G)\to \{1,\dots ,k\}$ has a monochromatic ordinary copy of $H$: a subgraph isomorphic to $H$ whose vertices all have one colour. The induced class ${\mathcal{R}}_{\mathrm{i}\mathrm{n}\mathrm{d}}^k(H)$ uses the same condition with induced copies, which also preserve nonadjacency. Equality of classes means equality of membership for every such graph $G$.

Graph and colouring conventions

Graph isomorphisms, automorphisms and graph complements

Ordinary vertex-Ramsey class

Induced vertex-Ramsey class

The article citation attributes the combined theorem. The argument above composes the separate ordinary and induced results.