Diagonal forms and zero-sum (mod 2) bipartite Ramsey numbers

Let G   be a subgraph of a complete bipartite graph Kn,nKn,n. Let N(G)N(G) be a 0-1 incidence matrix with edges of Kn,nKn,n against images of G   under the automorphism group of Kn,nKn,n. A diagonal form of N(G)N(G) is found for every G  , and the question as to whether the row space of N(G)N(G) over ZpZp contains the vector of all 1's is settled. This implies a new proof of Caro and Yuster's results on zero-sum bipartite Ramsey numbers, and provides necessary and sufficient conditions for the existence of a signed bipartite graph design.