A Deza-Frankl Type Theorem for Set Partitions

A set partition of [n] is a collection of pairwise disjoint nonempty subsets (called blocks) of [n] whose union is [n]. Let B(n) denote the family of all set partitions of [n]. A family A ⊆ B(n) is said to be m-intersecting if any two of its members have at least m blocks in common. For any set partition P ∈ B(n), let τ(P ) = {x : {x} ∈ P} denote the union of its singletons. Also, let μ(P ) = [n] − τ(P ) denote the set of elements that do not appear as a singleton in P . Let F2t = {P ∈ B(n) : |μ(P )| 6 t} ; F2t+1(i0) = {P ∈ B(n) : |μ(P ) ∩ ([n] \ {i0})| 6 t} . In this paper, we show that for r > 3, there exists a constant n0 = n0(r) depending on r such that for all n > n0, if A ⊆ B(n) is (n− r)-intersecting, then |A| 6 { |F2t|, if r = 2t; |F2t+1(1)|, if r = 2t+ 1. Moreover, equality holds if and only if A = { F2t, if r = 2t; F2t+1(i0), if r = 2t+ 1, for some i0 ∈ [n].