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 $n\ge Ak<sup>3$, for every fixed $A&gt;24$ and all sufficiently large $k$, uniformly over $1\le t&lt;k$. We also give counterexamples to Borg's conjecture at every intersection scale: for all sufficiently large $k$ and for every $1\le d&lt;k$, one may choose $d/4\le t\le d$ and $n=\lfloor tk<sup>2/3\rfloor$ so that a $t$-intersecting family is strictly larger than the canonical star.