Matching-star size Ramsey numbers under connectivity constraint

Recently, Caro, Patk\'os, and Tuza (2022) introduced the concept of connected Tur\'an number. We study a similar parameter in Ramsey theory. Given two graphs G₁ and G₂, the size Ramsey number r (G₁, G₂) refers to the smallest number of edges in a graph G such that for any red-blue edge-coloring of G, either a red subgraph G₁ or a blue subgraph G₂ is present in G. If we further restrict the host graph G to be connected, we obtain the connected size Ramsey number, denoted as rc (G₁, G₂). Erdos and Faudree (1984) proved that r (nK₂, K₁, ₌) =mn for all positive integers m, n. In this paper, we concentrate on the connected analog of this result. Rahadjeng, Baskoro, and Assiyatun (2016) provided the exact values of rc (nK₂, K₁, ₌) for n=2, 3. We establish a more general result: for all positive integers m and n with m (n²+2pn+n-3) /2, we have rc (nK₁, ₏, K₁, ₌) =n (m+p) -1. As a corollary, rc (nK₂, K₁, ₌) =nm+n-1 for m (n²+3n-3) /2. We also propose a conjecture for the interested reader.