Erd\H os--Ko--Rado type results for partitions via spread approximations

Abstract

In this paper, we address several Erd\H os--Ko--Rado type questions for families of partitions. Two partitions of $[n]$ are {\it $t$-intersecting} if they share at least $t$ parts, and are {\it partially $t$-intersecting} if some of their parts intersect in at least $t$ elements. The question of what is the largest family of pairwise $t$-intersecting partitions was studied for several classes of partitions: Peter Erd\H os and Sz\'ekely studied partitions of $[n]$ into $\ell$ parts of unrestricted size; Ku and Renshaw studied unrestricted partitions of $[n]$; Meagher and Moura, and then Godsil and Meagher studied partitions into $\ell$ parts of equal size. We improve and generalize the results proved by these authors. Meagher and Moura, following the work of Erd\H os and Sz\'ekely, introduced the notion of partially $t$-intersecting partitions, and conjectured, what should be the largest partially $t$-intersecting family of partitions into $\ell$ parts of equal size $k$. The main result of this paper is the proof of their conjecture for all $t, k$, provided $\ell$ is sufficiently large. All our results are applications of the spread approximation technique, introduced by Zakharov and the author. In order to use it, we need to refine some of the theorems from the original paper. As a byproduct, this makes the present paper a self-contained presentation of the spread approximation technique for $t$-intersecting problems.