Remarks on the size Ramsey number of graphs

In this paper we prove that the cyclomatic number of a graph whose every 2-edgecolouring contains a monochromatic path witht edges is not less than 3t/4 ? 2. This fact leads to a simple non-probabilistic proof of the following theorem of Beck: $$\begin{array}{*{20}c} {lim inf{{\hat r\left( {P_t } \right)} \mathord{\left/ {\vphantom {{\hat r\left( {P_t } \right)} t}} \right. \kern-\nulldelimiterspace} t} \geqslant {9 \mathord{\left/ {\vphantom {9 4}} \right. \kern-\nulldelimiterspace} 4},} & {t \to \infty ,} \\ \end{array}$$ where \(\hat r(P_t )\) is the size Ramsey number of a pathP _t ont edges. We also show that the size Ramsey number of a (q + 1)-edge star with a tail of length one equals 4q ? 2, i.e., it is linear on the number of edges of the graph. Finally, we calculate that the upper bound for the size Ramsey number of a (q + 2)-edge star with a tail of length two is not greater than 5q + 3.