Erdős-Ko-Rado theorem and Hilton-Milner type theorem for $k$-partitions

TL;DR

This work extends the Erdős–Ko–Rado paradigm to families of $k$-partitions of $[n]$ by studying $t$-intersecting partitions and introducing the $t$-cover concept. It proves that for $n\ge L(k,t)=(t+1)+(k-t+1)\log_2(t+1)(k-t+1)$, the largest $t$-intersecting family is trivial, consisting of all $k$-partitions containing $t$ fixed singletons, with size $\genfrac{\{}{\}}{0pt}{}{n-t}{k-t}$. For the stronger regime $n\ge 2L(k,t)$, it fully classifies the maximal non-trivial $t$-intersecting families as isomorphic to the natural candidates $\mathcal{A}(n,k,t)$ and $\mathcal{H}(n,k,t)$ (and $\mathcal{H}_1(n,k,t)$ in a range), yielding Hilton–Milner-type structure theorems for partition systems and enabling stability results. The analysis combines a $t$-cover framework with Stirling-number bounds and monotonicity lemmas, improving previous bounds and linking to classical partition-intersection results.

Abstract

A $k$-partition of an $n$-set $X$ is a collection of $k$ pairwise disjoint non-empty subsets whose union is $X$. A family of $k$-partitions of $X$ is called $t$-intersecting if any two of its members share at least $t$ blocks. A $t$-intersecting family is trivial if every $k$-partition in it contains $t$ fixed blocks, and is non-trivial otherwise. In this paper, we first prove that, for $n\geq L(k,t):=(t+1)+(k-t+1)\cdot\log_2(t+1)(k-t+1)$, a $t$-intersecting family with maximum size must consist of all $k$-partitions containing $t$ fixed singletons. This improves the results given by Erdős and Székely (2000), and by Kupavskii (2023). We further determine the non-trivial $t$-intersecting families of $k$-partitions with maximum size for $n \ge 2L(k,t)$, which turn out to be natural analogs of the corresponding families for finite sets. In addition, we prove a stability result.

Theorems & Definitions (22)

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