Erds-Hajnal Conjecture for Graphs with Bounded VC-Dimension

The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least e(logn)1-o(1). The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, ec<mml:msqrt>logn</mml:msqrt>, due to Erds and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial hereditary property. Our result is almost optimal and nearly matches the celebrated Erds-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least e(logn). Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties. Our main tool is a partitioning result found by Lovasz-Szegedy and Alon-Fischer-Newman, which is called the ultra-strong regularity lemma for graphs with bounded VC-dimension. We extend this lemma to k-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be (1/epsilon)O(d), improving the original bound of (1/epsilon)O(d2) in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an O(nk)-time algorithm for finding a partition meeting the requirements. Finally, we establish tight bounds on Ramsey-Turan numbers for graphs with bounded VC-dimension.