An upper bound for the size of a k-uniform intersecting family with covering number k

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• 作者：Andrii Arman^a ; ^{armana@myumanitoba.ca} ; Troy Retter^b ; ¹ ; ^{tretter@emory.edu}
• 关键词：Intersecting family ; Intersecting hypergraph ; Covering number ; Blocking number ; Transversal
• 刊名：Journal of Combinatorial Theory, Series A
• 出版年：2017
• 出版时间：April 2017
• 年：2017
• 卷：147
• 期：Complete
• 页码：18-26
• 全文大小：276 K
• 卷排序：147

文摘

Let r(k)r(k) denote the maximum number of edges in a k-uniform intersecting family with covering number k  . Erdős and Lovász proved that ⌊k!(e−1)⌋≤r(k)≤kk⌊k!(e−1)⌋≤r(k)≤kk. Frankl, Ota, and Tokushige improved the lower bound to r(k)≥(k/2)k−1r(k)≥(k/2)k−1, and Tuza improved the upper bound to r(k)≤(1−e−1+o(1))kkr(k)≤(1−e−1+o(1))kk. We establish that r(k)≤(1+o(1))kk−1r(k)≤(1+o(1))kk−1.