Edge-Ramsey class terminology in the motivation

PDF p. 1: Introduction, paragraph following Theorem 1.1

Let $H$ and $G$ be finite simple graphs with at least one vertex, and let $k$ be a positive integer. A $k$-edge-colouring of $G$ is any function $c:E(G)\to \{1,\dots ,k\}$; it need not use every colour. An ordinary copy of $H$ in $G$ is a subgraph of $G$ isomorphic to $H$, so extra edges of $G$ between its vertices are allowed. Such a copy is edge-monochromatic when $c(e)=c(f)$ for every two of its edges $e,f$; a copy with no edges is therefore edge-monochromatic.

The $k$-colour edge-Ramsey class of $H$ consists of those graphs $G$ for which every $k$-edge-colouring of $G$ has an edge-monochromatic ordinary copy of $H$. Here class membership is a predicate on graphs, with no choice of a universal set of labelled graphs implied.

Graph and colouring conventions

Unrestricted finite colourings and used palettes

Ordinary embeddings as injections preserving edges

The graph conventions specify the finite vertex and edge sets and the ordinary-copy terminology. The colouring definition applies to the finite set $E(G)$, including when it is empty, and gives a definite equality predicate on each copy's edges. The ordinary-embedding definition identifies a copy with the image of an injective edge-preserving map; it does not require nonedges of $H$ to remain nonedges in $G$. Consequently the displayed universal condition gives an unambiguous membership predicate, including for an edgeless $H$.

The article mentions edge Ramsey classes but does not specify a colour count in the cited question. The parameter $k$ makes the edge-colouring analogue explicit without asserting which parameter the cited question intends.