In this paper, we address several Erd\H os--Ko--Rado type questions for families of partitions. Two partitions of $[n]$ are $t$-intersecting if they share at least $t$ parts, and are 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 $k$. In this paper, we prove their conjecture for all $t, k$ and $\ell$ 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 their paper. As a byproduct, this makes the present paper a self-contained presentation of the spread approximation technique for $t$-intersecting problems.

翻译：本文研究划分族中的若干Erdős–Ko–Rado类型问题。两个$[n]$的划分称为$t$-相交的，若它们共享至少$t$个部分；称为部分$t$-相交的，若其中某些部分在至少$t$个元素上相交。针对若干划分类别，最大两两$t$-相交划分族的问题已被研究：Peter Erdős与Székely研究了$[n]$划分为大小无限制的$\ell$个部分的情形；Ku与Renshaw研究了$[n]$的无限制划分；Meagher与Moura，以及后续Godsil与Meagher研究了$[n]$划分为大小相等的$\ell$个部分的情形。本文改进并推广了上述作者的结果。继Erdős与Székely的工作之后，Meagher与Moura提出了部分$t$-相交划分的概念，并猜想：将$[n]$划分为大小均为$k$的$\ell$个部分时，最大的部分$t$-相交划分族应为何种形式。本文证明该猜想对所有充分大的$t$、$k$和$\ell$成立。所有结果均为Zakharov与本文作者提出的散布逼近方法的应用。为应用该方法，我们需对其论文中的若干定理进行精炼。作为副产品，本文也成为针对$t$-相交问题的散布逼近方法的自包含论述。