In this chapter we will study the weakest kind of this dependence – the members are required to be non-disjoint. A family is intersecting if any two of its sets have a non-empty intersection.doi:10.1007/978-3-642-17364-6_7Prof. Dr. Stasys Jukna...
The Hilton-Milner Theorem [8] is equivalent to finding the maximum size of an intersecting family with diversity at least 1. It is natural to ask the following question: given an intersecting family F of k-subset of [n], what is the maximum diversity it can possibly have? Let Fx={F:...
But this is impossible since, by (ii),Fcannot contain the set Btogether with its complement.To prove the second claim, take an arbitrary intersecting family and extendit to an ultraf i lter as follows. If there are some sets not in the family andsuch that their addition does not ...
A family \mathcal F\subset 2^{[n]} \mathcal F\subset 2^{[n]} is called intersecting if any two of its sets intersect. Given an intersecting family, its diversity is the number of sets not passing through the most popular element of the ground set. Peter Frankl made the following co...
Two classical problems in economics, the existence of a market equilibrium and the existence of social choice functions, are formalized here by the properties of a family of cones associated with the economy. It was recently established that a necessary and sufficient condition for solving the ...
A family of sets is said to be intersecting if A B for all A, B . It is a well-known and simple fact that an intersecting family of subsets of [n] = {1, 2, . . ., n} can contain at most 2n1 sets. Katona, Katona and Katona ask the following question. Suppose instead [n...
The Erdös-Ko-Rado theorem states that if F is a family of k-subsets of an n-set no two of which are disjoint, n ⩾ 2k, then |F| ⩽n−1k−1 holds. Taking all k-subsets through a point shows that this bound is best possible. Hilton and Milner showed that if ∩ F = ...
A family ${\\\cal F}$ of sets is said to be ( strictly ) EKR if no non-trivial intersecting sub-family of ${\\\cal F}$ is (as large as) larger than som... P Borg - 《Electronic Journal of Combinatorics》 被引量: 51发表: 2007年 The maximum size of a non-trivial intersectin...
One of the central notions is that of a r -wise t -intersecting family, that is, a collection F 1 ,…, F m of distinct subsets of the n -element set X such that | F i l ψ …ψF i r | ≥ t holds for all choices of 1≤ i l <…< i r ≤ m . What is the maximal...
Let A be a non-empty family of a-subsets of an n-element set and B a non-empty family of b-subsets satisfying A∩ B ≠ ⊘ for all A∈ A, B∈ B. Suppose that n⩾a+b, b⩾a. It is proved that in this case |A|+|B|⩽(nb)−n−ab) holds. Various extensions of...