Weak hypergraph regularity and applications to geometric Ramsey theory

Abstract

Let Δ = Δ 1 × … × Δ d ⊆ R n \Delta =\Delta _1\times \ldots \times \Delta _d\subseteq \mathbb {R}^n , where R n = R n 1 × ⋯ × R n d \mathbb {R}^n=\mathbb {R}^{n_1}\times \cdots \times \mathbb {R}^{n_d} with each Δ i ⊆ R n i \Delta _i\subseteq \mathbb {R}^{n_i} a non-degenerate simplex of n i n_i points. We prove that any set S ⊆ R n S\subseteq \mathbb {R}^n , with n = n 1 + ⋯ + n d n=n_1+\cdots +n_d of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of the configuration Δ \Delta . In particular any such set S ⊆ R 2 d S\subseteq \mathbb {R}^{2d} contains a d d -dimensional cube of side length λ \lambda , for all λ ≥ λ 0 ( S ) \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.