An analogue of the Hilton-Milner theorem for set partitions

摘要

Let B(n) denote the collection of all set partitions of [n]. Suppose A?B(n) is a non-trivial t-intersecting family of set partitions i.e. any two members of A have at least t blocks in common, but there is no fixed set of t blocks of size one which belong to all of them. It is proved that for sufficiently large n depending on t,|A|≤B_(n-t)-B~_(n-t)-B~_(n-t-1)+t where B _n is the n-th Bell number and B~n is the number of set partitions of [n] without blocks of size one. Moreover, equality holds if and only if A is equivalent to{P∈B(n):{1},{2},...,{t},{i}∈Pfor somei?{1,2,...,t,n}}∪{Q(i,n):1≤i≤t} where Q(i, n) = {{i, n}} ∪ {{j}: j ∈ [n] {set minus} {i, n}}. This is an analogue of the Hilton-Milner theorem for set partitions.