Nowhere dense Ramsey sets

A set of points S in Euclidean space Rᵈ is called Ramsey if any finite partition of R∞ yields a monochromatic copy of S. While characterization of Ramsey set remains a major open problem in the area, a stronger ``density'' concept was considered in [J. Amer. Math. Soc. 3, 1--7, 1990]: If S is a d-dimensional simplex, then for any μ>0 there is an integer d:=d(S,μ) and finite configuration X⊆ Rᵈ such that any subconfiguration Y⊆ X with |Y|≥ μ |X| contains a copy of S. Complementing this, here we show the existence of μ:=μ(S) and of an infinite configuration X⊆ R∞ with the property that any finite coloring of X yields a monochromatic copy of S, yet for any finite set of points Y⊆ X contains a subset Z⊆ Y of size |Z|≥ μ |Y| without a copy of S.