Weak hypergraph regularity and applications to geometric Ramsey theory. (English) Zbl 1529.11018
Summary: 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.
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