New values for the bipartite Ramsey number of the four-cycle versus stars

We provide new values of the bipartite Ramsey number RB(C4,K1,n) using induced subgraphs of the incidence graph of a projective plane. The approach, based on deleting subplanes of projective planes, has been used in related extremal problems and allows us to unify previous results and extend them. More importantly, using deep stability results on 2modp sets and double blocking sets, we can show some of the limits of this technique when the projective plane is Desarguesian of large enough square order. Finally, we also disprove two conjectures about RB(C4,K1,n).