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: $$begin{array}{*{20}c} {lim inf{{hat rleft( {P_t } right)} mathord{left/ {vphantom {{hat rleft( {P_t } right)} t}} right. kern-nulldelimiterspace} t} geqslant {9 mathord{left/ {vphantom {9 4}} right. kern-nulldelimiterspace} 4},} & {t to infty ,} end{array}$$ 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.