Intersecting -uniform families containing a given family

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• 作者：Jun Wang^a ; ^{jwang@shnu.edu.cn} ; Huajun Zhang^b ; ^{huajunzhang@zjnu.cn}
• 关键词：Family of sets ; Intersecting family ; Erdős&ndash ; Ko&ndash ; Rado theorem ; Hilton&ndash ; Milner theorem
• 刊名：Discrete Mathematics
• 出版年：2017
• 出版时间：6 February 2017
• 年：2017
• 卷：340
• 期：2
• 页码：140-144
• 全文大小：393 K
• 卷排序：340

文摘

A family AA of sets is said to be intersecting if A∩B≠∅A∩B≠∅ for any A,B∈AA,B∈A. Let m,n,km,n,k and rr be positive integers with m≥2k≥2r>n≥km≥2k≥2r>n≥k. A family FF of sets is called an (m,n,k,r)(m,n,k,r)-intersecting family   if FF is an intersecting subfamily of [m]k containing {A∈[m]k:|A∩[n]|≥r}. Maximum (m,k,k,k)(m,k,k,k)- and (m,k+1,k,k)(m,k+1,k,k)-intersecting families were determined by the well-known Erdős–Ko–Rado theorem and Hilton–Milner theorem, respectively. Recently, Li et al. determined maximum (m,n,k,k)(m,n,k,k)-intersecting family when n=2k−1,2k−2,2k−3n=2k−1,2k−2,2k−3 or mm is sufficiently large. In this paper, we determine all the maximum (m,n,k,r)(m,n,k,r)-intersecting families.