A degree version of the Hilton-Milner theorem

JOURNAL OF COMBINATORIAL THEORY SERIES A

卷:155

Article

Frankl, Peter¹  Han, Jie²  Huang, Hao³  Zhao, Yi⁴ 

[1] Alfred Renyi Inst Math, POB 127, H-1364 Budapest, Hungary

[2] Univ Sao Paulo, Inst Matemat & Estat, Rua Matao 1010, BR-05508090 Sao Paulo, Brazil

[3] Emory Univ, Dept Math & Comp Sci, Atlanta, GA 30322 USA

[4] Georgia State Univ, Dept Math & Stat, Atlanta, GA 30303 USA

关键词: Intersecting families;    Hilton-Milner theorem;    Erdds-Ko-Rado theorem;   

DOI  :  10.1016/j.jcta.2017.11.019

来源: Elsevier

【 摘 要 】

An intersecting family of sets is trivial if all of its members share a common element. Hilton and Milner proved a strong stability result for the celebrated Erdos-Ko-Rado theorem: when n > 2k, every non-trivial intersecting family of k-subsets of [n] has at most (n-1k-1) - (n-k-1 k-1) +1 members. One extremal family HMn,k consists of a k-set S and all k-subsets of [n] containing a fixed element x is an element of S and at least one element of S. We prove a degree version of the Hilton-Milner theorem: if n = Omega(k(2)) and F is a non-trivial intersecting family of k-subsets of [n], then delta(F) <= (HMn,k), where delta(F) denotes the minimum (vertex) degree of F. Our proof uses several fundamental results in extremal set theory, the concept of kernels, and a new variant of the Erdos-Ko-Rado theorem. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

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