On a conjecture of Erdős on size Ramsey number of star forests

Given two graphs F1 and F2, their size Ramsey number, denoted by r̂(F1, F2), is the minimum number of edges of a graph G such that for any edge coloring of G by colors red and blue, G contains either a red copy of F1 or a blue copy of F2. In this paper, we deal with the size Ramsey number of star forests (disjoint union of stars) and following a conjecture by Burr, Erdős, Faudree, Rousseau, and Schelp in 1978, we determine the exact value of r̂(⊔i = 1sK1, ni, ⊔i = 1tK1, mi) in several cases including when either mi’s and ni’s are odd, or s = 1 or s = 2 and n1 = n2.