Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5775136 | Journal of Mathematical Analysis and Applications | 2017 | 6 Pages |
Abstract
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|â¥Ï>0. Moreover, suppose that there exists 0<γ<â such that |Rj|/|Rk|â¤Î³ 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)â¥(NÏ(1+2â
γ1/n)n)1/2. If the sets in the collection {Rj} are convex, we obtain the improved Ramsey estimate R(N)â¥(3ânÏN)1/2. Applications of these results to weak type bounds of geometric maximal operators are provided.
Related Topics
Physical Sciences and Engineering
Mathematics
Analysis
Authors
P. Hagelstein, D. Herden, D. Young,