Ramsey-type theorems for sets satisfying a geometric regularity condition

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• 作者：P. Hagelstein ; ¹ ; ^{paul_hagelstein@baylor.edu" class="auth_mail" title="E-mail the corresponding author} ; D. Herden ^{daniel_herden@baylor.edu" class="auth_mail" title="E-mail the corresponding author} ; D. Young² ; ^{daniel_young1@baylor.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Ramsey theory ; Regularity condition ; Maximal operators
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 March 2017
• 年：2017
• 卷：447
• 期：2
• 页码：951-956
• 全文大小：291 K

文摘

We consider Ramsey-type problems associated to collections of sets in e7e7640c55e897ae" title="Click to view the MathML source">Rⁿ${\mathbb{R}}^n$ satisfying a standard geometric regularity condition. In particular, let e81650a7453b0d925f4c039ba4f86">${\{R_j\}}_{j=1}^N$ be a collection of measurable sets in e7e7640c55e897ae" title="Click to view the MathML source">Rⁿ${\mathbb{R}}^n$ such that every R_j$R_j$ is contained in a cube 9c098ef77fbe193ed88030ec2c2" title="Click to view the MathML source">Q_j$Q_j$ whose sides are parallel to the axes and such that 947940fdc01160c90812b10" title="Click to view the MathML source">|R_j|/|Q_j|≥ρ>0$|R_j|/|Q_j|\ge \rho >0$. Moreover, suppose that there exists 9483cae1ce281a773" title="Click to view the MathML source">0<γ<∞$0<\gamma <\infty$ such that |R_j|/|R_k|≤γ$|R_j|/|R_k|\le \gamma$ for every e850759c100c069abfee" title="Click to view the MathML source">j,k$j,k$. We prove that there exists a subcollection of e81650a7453b0d925f4c039ba4f86">${\{R_j\}}_{j=1}^N$ consisting of at least 8377980c227587eda730175783" title="Click to view the MathML source">R(N)$R(N)$ sets that either have a point of common intersection or that are pairwise disjoint, where 9893e8fad4e53e382d1f">$R(N)\ge {\left(\frac{N\rho }{{(1+2\cdot {\gamma }^{1/n})}^n}\right)}^{1/2}$. If the sets in the collection {R_j}$\{R_j\}$ are convex, we obtain the improved Ramsey estimate R(N)≥(3⁻ⁿρN)^1/2$R(N)\ge {\left(3^{-n}\rho N\right)}^{1/2}$. Applications of these results to weak type bounds of geometric maximal operators are provided.