### The connected size Ramsey number for matchings versus small disconnected graphs

#### Abstract

Let *F*, *G*, and *H* be simple graphs. The notation *F* → (*G*, *H*) means that if all the edges of *F* are arbitrarily colored by red or blue, then there always exists either a red subgraph *G* or a blue subgraph *H*. The size Ramsey number of graph *G* and *H*, denoted by *r̂*(*G*, *H*) is the smallest integer *k* such that there is a graph *F* with *k* edges satisfying *F* → (*G*, *H*). In this research, we will study a modified size Ramsey number, namely the connected size Ramsey number. In this case, we only consider connected graphs *F* satisfying the above properties. This connected size Ramsey number of *G* and *H* is denoted by *r̂*_{c}(*G*, *H*). We will derive an upper bound of *r̂*_{c}(*n**K*_{2}, *H*), *n* ≥ 2 where *H* is 2*P*_{m} or 2*K*_{1, t}, and find the exact values of *r̂*_{c}(*n**K*_{2}, *H*), for some fixed *n*.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2019.7.1.9

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ISSN: 2338-2287

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