Further asymptotic size Ramsey results obtained via linear programming

Recently, the author (SIAM J. Discrete Math. 16 (2003) 99-113) has asymptotically computed (via linear programming) size Ramsey numbers involving complete bipartite graphs. Here an attempt is made to extend this method to a larger class of problems by considering the 'simplest' open case when one of the forbidden graphs is S-1,S-n (the n-star K-1,K-n with an added leaf). Although we obtain new non-trivial results such as, for example, (r) over cap (K2,nS1,n) = (9 + o(1))n and (r) over cap (K-3,K-n,S-1,S-n)= (16 + o(1))n, even this 'simple' case remains open. (C) 2003 Elsevier B.V. All rights reserved.