Complete Kneser transversals

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• 作者：J. Chappelon^a ; ^{jonathan.chappelon@umontpellier.fr" class="auth_mail" title="E-mail the corresponding author} ; L. Martí ; nez-Sandoval^b ; ^{leomtz@im.unam.mx" class="auth_mail" title="E-mail the corresponding author} ; L. Montejano^b ; ^{luis@math.unam.mx" class="auth_mail" title="E-mail the corresponding author} ; L.P. Montejano^a ; ^{lpmontejano@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; J.L. Ramí ; rez Alfonsí ; n^a ; ^{jorge.ramirez-alfonsin@umontpellier.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：52A35 ; 05C15 ; 52C40 ; 52B40 ; 52B99
• 刊名：Advances in Applied Mathematics
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：82
• 期：Complete
• 页码：83-101
• 全文大小：401 K

文摘

Let b0bfd2bea6d537aebb" title="Click to view the MathML source">k,d,λ⩾1$k,d,\lambda ⩾1$ be integers with bd" title="Click to view the MathML source">d⩾λ$d⩾\lambda$. Let e431fb952f126b9c482d717585b90e6" title="Click to view the MathML source">m(k,d,λ)$m(k,d,\lambda )$ be the maximum positive integer n such that every set of n   points (not necessarily in general position) in a3cb77fd3ee448" title="Click to view the MathML source">R^d${\mathbb{R}}^d$ has the property that the convex hulls of all k  -sets have a common transversal bd" title="Click to view the MathML source">(d−λ)$(d-\lambda )$-plane. It turns out that e431fb952f126b9c482d717585b90e6" title="Click to view the MathML source">m(k,d,λ)$m(k,d,\lambda )$ is strongly connected with other interesting problems, for instance, the chromatic number of Kneser hypergraphs and a discrete version of Rado's centerpoint theorem. In the same spirit, we introduce a natural discrete version 9363b081ed7fec36478a69566ee1" title="Click to view the MathML source">m^⁎$m^{⁎}$ of m by considering the existence of complete Kneser transversals  . We study the relation between them and give a number of lower and upper bounds of 9363b081ed7fec36478a69566ee1" title="Click to view the MathML source">m^⁎$m^{⁎}$ as well as the exact value in some cases. The main ingredient for the proofs are Radon's partition theorem as well as oriented matroids tools. By studying the alternating oriented matroid we obtain the asymptotic behavior of the function 9363b081ed7fec36478a69566ee1" title="Click to view the MathML source">m^⁎$m^{⁎}$ for the family of cyclic polytopes.