Ks,tKs,t-saturated bipartite graphs

An nn-by-nn bipartite graph is HH-saturated if the addition of any missing edge between its two parts creates a new copy of HH. In 1964, Erdős, Hajnal and Moon made a conjecture on the minimum number of edges in a Ks,sKs,s-saturated bipartite graph. This conjecture was proved independently by Wessel and Bollobás in a more general, but ordered, setting: they showed that the minimum number of edges in a K(s,t)K(s,t)-saturated bipartite graph is n2−(n−s+1)(n−t+1)n2−(n−s+1)(n−t+1), where K(s,t)K(s,t) is the “ordered” complete bipartite graph with ss vertices in the first color class and tt vertices in the second. However, the very natural question of determining the minimum number of edges in the unordered Ks,tKs,t-saturated case remained unsolved. This problem was considered recently by Moshkovitz and Shapira who also conjectured what its answer should be. In this short paper we give an asymptotically tight bound on the minimum number of edges in a Ks,tKs,t-saturated bipartite graph, which is only smaller by an additive constant than the conjecture of Moshkovitz and Shapira. We also prove their conjecture for K2,3K2,3-saturation, which was the first open case.