The size Ramsey number ^r(F1; F2) is the smallest number of edges that an (F 1; F2)-arrowing graph can have. Let Si;n be obtained fromthe star K 1;n by subdividing one edge by new i\Gamma 1 vertices. Bielak [Periodica Math. Hung. 18 (1987) 27-38] showed that ^r(S 1;n; S1;n) =4n \Gamma 2 and that ^r(S 2;n; S2;n)  ^ 5n + 3. We compute asymptotically all unknown values of ^r(S _;n; S*;n), 0 ^  _  ^  *  ^ 2. In particular, we show that ^r(S2;n; S2;n)  = 19 4 n + O(1)