Abstract
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 $$e^{(\log n)^{1 - o(1)}}$$ . The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, $$e^{c\sqrt{\log n}}$$ , due to Erdős 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 Erdős–Hajnal conjecture, according to which one can always find a clique or an independent set of size at least $$e^{\Omega (\log n)}$$ . 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 $$(1/\varepsilon )^{O(d)}$$ , improving the original bound of $$(1/\varepsilon )^{O(d^2)}$$ in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an $$O(n^k)$$ -time algorithm for finding a partition meeting the requirements. Finally, we establish tight bounds on Ramsey–Turan numbers for graphs with bounded VC-dimension.
Highlights
During the relatively short history of computational geometry, there were many breakthroughs that originated from results in extremal combinatorics [27]
The aim of this paper is to investigate whether the observation that the Erdos-Hajnal conjecture tends to hold for geometrically defined graphs can be ascribed to the fact that they have bounded
Many interesting results arose in our study of graphs and hypergraphs with bounded VC-dimension
Summary
During the relatively short history of computational geometry, there were many breakthroughs that originated from results in extremal combinatorics [27]. For fixed d ≥ 5 and for every sufficiently large n, there is an n-vertex graph G = (V, E) with VC-dimension at most d with the property that there are no two disjoint subsets A, B ⊂ V (G) such that |A|, |B| ≥ 4n4/d log n and (A, B) is homogeneous, that is, either A × B ⊂ E(G) or (A × B) ∩ E(G) = ∅ It follows from a result of Alon et al [3] that a stronger regularity lemma holds for semi-algebraic graphs of bounded description complexity, where all but an ε-fraction of the pairs of parts in the equitable partition are complete or empty, instead of just ε-homogeneous as in the bounded VCdimension case (see [37]).
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