Ramsey Numbers of Multiple Copies of Graphs in a Component

For a graph G, let \(R({\mathcal {C}}(nG))\) denote the least N such that every 2-colouring of the edges of \(K_N\) contains a monochromatic copy of nG in a monochromatic connected subgraph, where nG denotes n vertex disjoint copies of G. Gyárfás and Sárközy (J Graph Theory 83(2):109–119, 2016) showed that \(R({\mathcal {C}}(nK_3))=7n-2\) for \(n \ge 2\). After that, Roberts (Electron J Comb 24(1):8, 2017)showed that \(R({\mathcal {C}}(nK_r))=(r^2-r+1)n-r+1\) for \(r \ge 4\) and \(n \ge R(K_r)\), where \(R(K_r)\) is the Ramsey number of \(K_r\). In this paper, we determine \(R({\mathcal {C}}(nG))\) for all 4-vertex graphs G without isolated vertices.