Connected size Ramsey number for matchings vs. small stars or cycles

The notation F → ( G , H ) means that if the edges of F are colored red and blue, then the red subgraph contains a copy of G or the blue subgraph contains a copy of H . The connected size Ramsey number r ^ c ( G , H ) of graphs G and H is the minimum size of a connected graph F satisfying F → ( G , H ) . For m ≥ 2 , the graph consisting of m independent edges is called a matching and is denoted by m K 2 . In 1981, Erdös and Faudree determined the size Ramsey numbers for the pair ( m K 2 , K 1 , t ) . They showed that the disconnected graph m K 1 , t → ( m K 2 , K 1 , t ) for t , m ≥ 1 . In this paper, we will determine the connected size Ramsey number r ^ c ( n K 2 , K 1 , 3 ) for n ≥ 2 and r ^ c ( 3 K 2 , C 4 ) . We also derive an upper bound of the connected size Ramsey number r ^ c ( n K 2 , C 4 ) , for n ≥ 4 .