A Single-Exponential Erdős--Hajnal Bound for Graphs of Bounded VC-Dimension

A homogeneous set in a graph is a clique or a stable set. The Erd\H{o}s--Hajnal conjecture states that, for every graph $H$, there exists $c>0$ such that every $H$-free graph on $n$ vertices has a homogeneous set of size at least $n^c$. Nguyen, Scott and Seymour proved that for every $d>0$, graphs of VC-dimension at most $d$ have the Erd\H{o}s--Hajnal property, confirming a conjecture of Fox, Pach and Suk. In particular, they showed that every such $n$-vertex graph contains a homogeneous set of size at least $n^{\eta_d}$ for some $\eta_d\ge 2^{-2^{O(d)}}$. In this paper, we give a sharper quantitative bound on the homogeneous sets in graphs of VC-dimension at most $d$, showing that one may take $\eta_d\ge (Cd)^{-d},$ where $C$ is an absolute constant. Equivalently, every graph $G$ of VC-dimension at most $d$ satisfies $$\max\{\omega(G),\alpha(G)\}\ge |G|^{(Cd)^{-d}}.$$ Our proof refines the iterative sparsification method of Nguyen, Scott and Seymour. The main enhancement is to apply the VC-dimension assumption directly, which gives a more efficient induction and thus improves the dependence on $d$. We also derive quantitative consequences for polynomial R\"odl subgraphs, hypergraph Ramsey bounds under bounded VC-dimension, induced-free and viral formulations, tournaments, NIP and semi-algebraic graphs, Boolean combinations of relations of bounded VC-dimension, graphs whose adjacency matrices have bounded rank, graphs of bounded sign-rank, and graphs defined by dot-product threshold representations.