Weak hypergraph regularity and applications to geometric Ramsey theory

Abstract

Let $\Delta ={\Delta }_1\times \dots \times {\Delta }_d\subseteq {\mathbb{R}}^n$, where ${\mathbb{R}}^n={\mathbb{R}}^{n_1}\times \cdots \times {\mathbb{R}}^{n_d}$ with each ${\Delta }_i\subseteq {\mathbb{R}}^{n_i}$ a non-degenerate simplex of $n_i$ points. We prove that any set $S\subseteq {\mathbb{R}}^n$, with $n=n_1+\cdots +n_d$ of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of the configuration $\Delta$. In particular any such set $S\subseteq {\mathbb{R}}^{2d}$ contains a $d$-dimensional cube of side length $\lambda$, for all $\lambda \ge {\lambda }_0(S)$. We also prove analogous results with the underlying space being the integer lattice. The proof is based on a weak hypergraph regularity lemma and an associated counting lemma developed in the context of Euclidean spaces and the integer lattice.