#### Vol. 13, No. 3, 2020

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Distance graphs and sets of positive upper density in $\mathbb{R}^d$

### Neil Lyall and Ákos Magyar

Vol. 13 (2020), No. 3, 685–700
DOI: 10.2140/apde.2020.13.685
##### Abstract

We present a refinement and sharp extension of a result of Bourgain on finding configurations of $k+1$ points in general position in measurable subset of ${ℝ}^{d}$ of positive upper density whenever $d\ge k+1$ to all proper $k$-degenerate distance graphs.

##### Keywords
distance graphs, uniformity norms, geometric Ramsey theory
Primary: 11B30