A degree version of the Hilton-Milner theorem

Author(s)

Frankl, Peter ^[1] ; Han, Jie ^[2] ; Huang, Hao ^[3] ; Zhao, Yi ^[4]

Affiliation

^[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

• Document type: Journal article
• Source: JOURNAL OF COMBINATORIAL THEORY SERIES A; v. 155, p. 493-502, APR 2018.
• Web of Science Citations: 3

Abstract

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. (AU)

• FAPESP's process: 14/18641-5 - Hamilton cycles and tiling problems in hypergraphs
• Grantee: Jie Han
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• FAPESP's process: 13/03447-6 - Combinatorial structures, optimization, and algorithms in theoretical Computer Science
• Grantee: Carlos Eduardo Ferreira
• Support Opportunities: Research Projects - Thematic Grants