Generalised Ramsey Numbers with respect to Classes of Graphs

Let \(\mathcal{H}_1, \ldots, \mathcal{H}_t\) be classes of graphs. The class Ramsey number \(R(\mathcal{H}_1, \ldots, \mathcal{H}_t)\) is the smallest integer \(n\) such that for each \(t\)-edge colouring \((G_1, \ldots, G_t)\) of \(K_n\), there is at least one \(i \in \{1, \ldots, t\}\) such that \(G_i\) contains a subgraph \(H_i \in \mathcal{H}_i\). We take \(t = 2\) and determine \(R(\mathcal{G}^1_l, \mathcal{G}^1_m)\) for all \(2 \leq l \leq m\) and \(R(\mathcal{G}^2_i, \mathcal{G}^2_{m})\) for all \(3 \leq l \leq m\), where \(\mathcal{G}^i_j\) consists of all edge-minimal graphs of order \(j\) and minimum degree \(i\).