Erd\H{o}s-Ko-Rado theorem and Hilton-Milner type theorem for $k$-partitions

TL;DR

This paper extends classical intersection theorems to $k$-partitions of sets, identifying maximum families and their structures for large $n$, and provides stability results for these configurations.

Contribution

It proves new bounds and characterizations for maximum $t$-intersecting families of $k$-partitions, generalizing Erdős-Ko-Rado and Hilton-Milner theorems.

Findings

Maximum size $t$-intersecting families contain all partitions with $t$ fixed singletons.

Non-trivial maximum families are characterized for sufficiently large $n$.

Stability results show near-maximal families are close to the extremal structure.

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 Erd\H{o}s and Sz\'{e}kely (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.