Weak hypergraph regularity and applications to geometric Ramsey theory

Abstract:

Let $\Delta =\Delta _1\times \ldots \times \Delta _d\subseteq \mathbb {R}^n$, 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 $\lambda \geq \lambda _0(S)$. 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.