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

Let G be a subgraph of a complete bipartite graph Kn,n. Let N(G) be a 0-1 incidence matrix with edges of Kn,n against images of G under the automorphism group of Kn,n. A diagonal form of N(G) is found for every G, and whether the row space of N(G) over Zp contains the vector of all 1’s is determined. This re-proves Caro and Yuster’s results on zero-sum bipartite Ramsey numbers [3], and provides necessary and sufficient conditions for the existence of a signed bipartite graph design.