Constructing trees in bipartite graphs

Abstract In this paper we shall show that if G = ( V , E ) is a bipartite graph with more than ( a − 1)‖ Y ‖ + ( b − 1)‖ X ‖ − ( a − 1)( b − 1) edges, where ( X , Y ) is a vertex-partition for G and a ⩽ b are natural numbers with a ⩽ ‖ X ‖, b ⩽ ‖ Y ‖, then G contains every tree T with bipartite-numbers a ⩽ b . This result is related to Ramsey-theory for trees.