Simplex--center configurations in dense subsets of Euclidean spaces and the integer lattice

Abstract

We obtain density Ramsey theorems for configurations consisting of the vertices of a simplex ${\Delta }_o$ together with their barycenter. We prove that any subset $A\subseteq {\mathbb{R}}^n$ of positive upper density contains an isometric copy of all sufficiently large dilates of ${\Delta }_o$ together with its barycenter. As this configuration is non-spherical such results are not possible with respect to the quadratic Euclidean metric, we consider general metrics $\rho$ defined by a positive-definite, homogeneous forms of even degree at least four. We prove the analogous result in the discrete setting, for subsets $A$ of the integer lattice ${\mathbb{Z}}^n$, under some natural and necessary congruence restrictions on the scales $\lambda$ at which the set $A$ can contain an isometric copy of the simplex.