Intersecting integer partitions

Abstract

If $a_1, a_2, ..., a_k$ and $n$ are positive integers such that $n = a_1 + a_2 + ... + a_k$, then the sum $a_1 + a_2 + ... + a_k$ is said to be a \emph{partition of $n$} of \emph{length $k$}, and $a_1, a_2, ..., a_k$ are said to be the \emph{parts} of the partition. Two partitions that differ only in the order of their parts are considered to be the same. We say that two partitions \emph{intersect} if they have at least one common part. We call a set $A$ of partitions \emph{intersecting} if any two partitions in $A$ intersect. Let $P_{n,k}$ be the set of all partitions of $n$ of length $k$. We conjecture that if $2 \leq k \leq n$, then the size of any intersecting subset of $P_{n,k}$ is at most the size of $P_{n-1,k-1}$, which is the size of the intersecting subset of $P_{n,k}$ consisting of those partitions which have 1 as a part. The conjecture is trivially true for $n \leq 2k$, and we prove it for $n \geq 5k^5$. We also generalise this for subsets of $P_{n,k}$ with the property that any two of their members have at least $t$ common parts.