Large homogeneous subgraphs in bipartite graphs with forbidden induced subgraphs

Abstract

For a bipartite graph G, let h similar to ( G ) be the largest t such that either G contains K t , t, a complete bipartite subgraph with parts of size t, or the bipartite complement of G contains K t , t as a subgraph. For a class of graphs F, let h similar to ( F ) = min { h similar to ( G ) : G is an element of F }. We say that a bipartite graph H is strongly acyclic if neither H nor its bipartite complement contains a cycle. By Forb ( n , H ) we denote the set of bipartite graphs with parts of size n, which do not contain H as an induced bipartite subgraph respecting the sides. One can easily show that h similar to ( Forb ( n , H ) ) = O ( n 1 - epsilon ) for a positive epsilon if H is not strongly acyclic. Here we ask whether h similar to ( Forb ( n , H ) ) is linear in n for any strongly acyclic graph H. We answer this question in the positive for all but four strongly acyclic graphs. We do not address this question for the remaining four graphs in this paper.