Intersecting integer partitions: star bounds and counterexamples at every scale

Abstract

Two integer partitions $t$-intersect if they have at least $t$ common parts, counted with multiplicity. We study the largest $t$-intersecting families of integer partitions of $n$ into exactly $k$ positive parts. The canonical $t$-star consists of the partitions containing at least $t$ ones. Applying Kupavskii's weak-spread theorem, we prove that this star is largest whenever $n\ge A{k}^3$, for every fixed $A>24$ and all sufficiently large $k$, uniformly over $1\le t<k$. We also give counterexamples to Borg's conjecture at every intersection scale: for all sufficiently large $k$ and for every $1\le d<k$, one may choose $d/4\le t\le d$ and $n=\lfloor t{k}^2/3\rfloor$ so that a $t$-intersecting family is strictly larger than the canonical star.