Non-trivial intersecting uniform sub-families of hereditary families

详细信息    查看全文

• 作者：Peter Borg ^{p.borg.02@cantab.net peter.borg@um.edu.mt}
• 关键词：Non-trivial intersecting families ; Hereditary families ; Erd?s&ndash ; Ko&ndash ; Rado theorem ; Hilton&ndash ; Milner theorem ; Extremal set theory
• 刊名：Discrete Mathematics
• 出版年：2013
• 出版时间：6 September, 2013
• 年：2013
• 卷：313
• 期：17
• 页码：1754-1761
• 全文大小：391 K

文摘

For a family of sets, let denote the size of a smallest set in that is not a subset of any other set in , and for any positive integer , let denote the family of -element sets in . We say that a family is of Hilton-Milner (HM) type if for some , all sets in have a common element and intersect . We show that if a hereditary family is compressed and , then the HM-type family is a largest non-trivial intersecting sub-family of ; this generalises a well-known result of Hilton and Milner. We demonstrate that for any and , there exist non-compressed hereditary families with such that no largest non-trivial intersecting sub-family of is of HM type, and we suggest two conjectures about the extremal structures for arbitrary hereditary families.