Pure pairs. IX. Transversal trees

Fix k>0, and let G be a graph, with vertex set partitioned into k subsets (`blocks') of approximately equal size. An induced subgraph of G is transversal (with respect to this partition) if it has exactly one vertex in each block (and therefore it has exactly k vertices). A pure pair in G is a pair X,Y of disjoint subsets of V(G) such that either all edges between X,Y are present or none are; and in the present context we are interested in pure pairs (X,Y) where each of X,Y is a subset of one of the blocks, and not the same block. This paper collects several results and open questions concerning how large a pure pair must be present if various types of transversal subgraphs are excluded.