2011/09/02 by Cheng Yeaw Ku, Ku, Cheng Yeaw, Kok Bin Wong +1
Mathematics · #Advanced Combinatorial Mathematics #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #math.CO
paper · pdf · doi:10.48550/arxiv.1109.0417
arxiv created 2011/09/02 · openalex publication_date 2011/09/02 · arxiv updated 2011/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 t blocks of size one which belong to all of them. It is proved that for sufficiently large n depending on t, |A| ≤ Bn-t-Bn-t-Bn-t-1+t where Bn is the n-th Bell number and Bn 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\ ∈ P \textnormalfor some i \not = 1,2,..., t,n \∪ \Q(i,n) : 1≤ i≤ t\ where Q(i,n)=\\i,n\\∪\\j\ : j∈ [n]∖ \i,n\\. This is an analogue of the Hilton-Milner theorem for set partitions.