Erdős-Ko-Rado theorems for uniform set-partition systems

Abstract

Two set partitions of an $n$-set are said to $t$-intersect if they have $t$ classes in common. A $k$-partition is a set partition with $k$ classes and a $k$-partition is said to be uniform if every class has the same cardinality $c=n/k$. In this paper, we prove a higher order generalization of the Erdős-Ko-Rado theorem for systems of pairwise $t$-intersecting uniform $k$-partitions of an $n$-set. We prove that for $n$ large enough, any such system contains at most $${1\over(k-t)!} {n-tc \choose c} {n-(t+1)c \choose c} \cdots {n-(k-1)c \choose c}$$ partitions and this bound is only attained by a trivially $t$-intersecting system. We also prove that for $t=1$, the result is valid for all $n$. We conclude with some conjectures on this and other types of intersecting partition systems.

An Erdős–Ko–Rado theorem for subset partitions

“…We shall require lemmas similar to the Lemma 3 used by Meagher and Moura in [11]-the proofs of which use similar counting arguments. As we shall see, it is worthwhile to consider the size of a canonical t-intersecting family of (k, ℓ)-subpartitions, and find when this is an upper bound for the size of any t-intersecting family of (k, ℓ)-subpartitions.…”

“…For the intersecting families being investigated here, each (k, ℓ)-subpartition in the family is also a dominating set. In [11], dominating sets were called blocking sets. We use the term dominating set here because if the classes in the (k, ℓ)-subpartitions (the k-sets) are considered to be vertices, then each (k, ℓ)subpartition can be thought of as an edge in an ℓ-uniform hypergraph on these vertices.…”

Erdős–Ko–Rado for perfect matchings

“…Since a perfect matching of K 2n can be seen as a n/2-uniform partition of [2n], a combinatorial proof Theorem 1.1 was first given by Meagher and Moura via the EKR theorem for intersecting families of k-uniform partitions [19]. The case where k = n/2 arises as a special case in their proof and is the most difficult part of their result.…”

An algebraic proof of the Erdős-Ko-Rado theorem for intersecting families of perfect matchings

“…The characterization of the maximum set of intersecting perfect matchings follows from the characterization of the facets of this polytope. Further, with this characterization, we are able to prove that the perfect matching derangement graph is not a Cayley graph.Meagher and Moura [10] proved a version of the EKR theorem holds for intersecting uniform partitions using a counting argument [10]. This result includes the EKR theorem for perfect matchings.…”

“…Meagher and Moura [10] proved a version of the EKR theorem holds for intersecting uniform partitions using a counting argument [10]. This result includes the EKR theorem for perfect matchings.…”