A note on Euclidean Ramsey theory (Q2385141)

If \(n\geq 2\), \(X\subseteq {\mathbb R}^n\) is infinite, \(Y\subseteq {\mathbb R}^n\) is an \((n+1)\)-element affine independent set, then there is a subset of \({\mathbb R}^n\) which hits every similar copy of \(X\) and does not contain a subset similar to \(Y\).