The size Ramsey and connected size Ramsey numbers for matchings versus paths. Issue 1 (January 2021) - Vito, V; Nabila, A C; Safitri, E; Silaban, D R
Abstract: Given simple graphs F, G, and H, we write F → ( G, H ) if for every red-blue coloring of the edges of F, there exists either a red subgraph G or a blue subgraph H in F . The size Ramsey number for G and H, denoted by r ^ ( G, H ), is the smallest size of a graph F which satisfies F → ( G, H ). If in addition F must be connected, then the resulting number is called the connected size Ramsey number for G and H, denoted by r ^ c ( G, H ). In this paper, we obtain upper bounds for r ^ ( t K 2, P m ), r ^ c ( t K 2, P m ), r ^ ( t K 2, P 4 +  a leaf  ), and r ^ c ( t K 2, P 4 +  a leaf  ) for t ≥ 1 and m ≥ 3. We also determine the exact values of r ^ c ( 2 K 2, P m ) and r ^ ( t K 2, P 3 ) . In addition, the exact values of r ^ c ( t K 2, P 3 ) for t = 3, 4 are given.