Ramsey-type theorems for sets satisfying a geometric regularity condition

We consider Ramsey-type problems associated to collections of sets in Rn satisfying a standard geometric regularity condition. In particular, let {Rj}j=1N be a collection of measurable sets in Rn such that every Rj is contained in a cube Qj whose sides are parallel to the axes and such that |Rj|/|Qj|≥ρ>0. Moreover, suppose that there exists 0<γ<∞ such that |Rj|/|Rk|≤γ for every j,k. We prove that there exists a subcollection of {Rj}j=1N consisting of at least R(N) sets that either have a point of common intersection or that are pairwise disjoint, where R(N)≥(Nρ(1+2⋅γ1/n)n)1/2. If the sets in the collection {Rj} are convex, we obtain the improved Ramsey estimate R(N)≥(3−nρN)1/2. Applications of these results to weak type bounds of geometric maximal operators are provided.