Erdos-Hajnal Properties For Powers Of Sparse Graphs

We prove that for every nowhere dense class of graphs C, positive integer d, and epsilon > 0, the following holds: in every n-vertex graph G from C one can find two disjoint vertex subsets A, B subset of V (G) such that vertical bar A vertical bar >= (1/2 - epsilon). n and vertical bar B vertical bar = Omega (n1-epsilon); and either dist(a, b) <= d for all a is an element of A and b is an element of B, or dist(a, b) > d for all a is an element of A and b is an element of B. We also show some stronger variants of this statement, including a generalization to the setting of first-order interpretations of nowhere dense graph classes.