An Analogue of Hilton-Milner Theorem for Set Partitions

Abstract: Let $\mathcal{B}(n)$ denote the collection of all set partitions of $[n]$. Suppose $\mathcal{A} \subseteq \mathcal{B}(n)$ 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 $t$, [ |\mathcal{A}| \le B_{n-t}-\tilde{B}{n-t}-\tilde{B}{n-t-1}+t ] where $B_{n}$ is the $n$-th Bell number and $\tilde{B}_{n}$ is the number of set partitions of $[n]$ without blocks of size one. Moreover, equality holds if and only if $\mathcal{A}$ is equivalent to [ {P \in \mathcal{B}(n): {1}, {2},..., {t}, {i} \in P \textnormal{for some} i \not = 1,2,..., t,n }\cup {Q(i,n)\ :\ 1\leq i\leq t} ] where $Q(i,n)={{i,n}}\cup{{j}\ :\ j\in [n]\setminus {i,n}}$. This is an analogue of the Hilton-Milner theorem for set partitions.