Size-Ramsey numbers of cycles versus a path

The size-Ramsey number Rˆ(F,H) of a family of graphs F and a graph H is the smallest integer m such that there exists a graph G on m edges with the property that any coloring of the edges of G with two colors, say, red and blue, yields a red copy of a graph from F or a blue copy of H. In this paper we first focus on F=C≤cn, where C≤cn is the family of cycles of length at most cn, and H=Pn. In particular, we show that 2.00365n≤Rˆ(C≤n,Pn)≤31n. Using similar techniques, we also managed to analyze Rˆ(Cn,Pn), which was investigated before but until last year only by using the regularity method.