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

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