88 research outputs found
Cross-intersecting families and primitivity of symmetric systems
Let be a finite set and , the power set of ,
satisfying three conditions: (a) is an ideal in , that is,
if and , then ; (b) For with , if for any
with ; (c) for every . The
pair is called a symmetric system if there is a group
transitively acting on and preserving the ideal . A
family is said to be a
cross--family of if for any and with . We prove that if is a
symmetric system and is a
cross--family of , then where . This generalizes Hilton's theorem on
cross-intersecting families of finite sets, and provides analogs for
cross--intersecting families of finite sets, finite vector spaces and
permutations, etc.
Moreover, the primitivity of symmetric systems is introduced to characterize
the optimal families.Comment: 15 page
Erdos-Ko-Rado from intersecting shadows
A set system is called t-intersecting if every two members meet each other in at least t elements. Katona determined the minimum ratio of the shadow and the size of such families and showed that the Erdos-Ko-Rado theorem immediately follows from this result. The aim of this note is to reproduce the proof to obtain a slight improvement in the Kneser graph. We also give a brief overview of corresponding results
Strongly intersecting integer partitions
We call a sum a1+a2+• • •+ak a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that a1 ≤ a2 ≤ • • • ≤ ak and n = a1 + a2 + • • • + ak. For i = 1, 2, . . . , k, we call ai the ith part of the sum a1 + a2 + • • • + ak. Let Pn,k be the set of all partitions of n of length k. We say that two partitions a1+a2+• • •+ak and b1+b2+• • •+bk strongly intersect if ai = bi for some i. We call a subset A of Pn,k strongly intersecting if every two partitions in A strongly intersect. Let Pn,k(1) be the set of all partitions in Pn,k whose first part is 1. We prove that if 2 ≤ k ≤ n, then Pn,k(1) is a largest strongly intersecting subset of Pn,k, and uniquely so if and only if k ≥ 4 or k = 3 ≤ n ̸∈ {6, 7, 8} or k = 2 ≤ n ≤ 3.peer-reviewe
A cross-intersection theorem for subsets of a set
Two families and of sets are said to be
cross-intersecting if each member of intersects each member of
. For any two integers and with , let
denote the family of all subsets of of
size at most . We show that if ,
, and and
are cross-intersecting, then and equality
holds if and
. Also, we
generalise this to any number of such cross-intersecting families.Comment: 12 pages, submitted. arXiv admin note: text overlap with
arXiv:1212.695
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