Most Probably Intersecting Families of Subsets

Let F be a family of subsets of an n-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the inequality vertical bar F vertical bar <= 2(n-1). Suppose that vertical bar F vertical bar = 2(n-1) + i. Choose the members of F independently with probability p (delete them with probability 1 - p). The new family is intersecting with a certain probability. We try to maximize this probability by choosing F appropriately. The exact maximum is determined in this paper for some small i. The analogous problem is considered for families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all [n/2]-element subsets. We determine the most probably inclusion-free family too, when the number of members is (n([n/2])) + 1

Constructing Union-Free pairs of K-Element subsets

It is proved that one can choose [1/2(n/k)] disjoint pairs of k-element subsets of an n-element set in such a way that the unions of the pairs are all different, supposing that n > n(k)

Color the cycles

The cycles of length k in a complete graph on n vertices are colored in such a way that edge-disjoint cycles get distinct colors. The minimum number of colors is asymptotically determined. © 2013

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

A general 2-part Erdős-Ko-Rado theorem

A two-part extension of the famous Erdo{combining double acute accent}s-Ko-Rado Theorem is proved. The underlying set is partitioned into X1 and X2. Some positive integers ki, ℓi (1 ≤ i ≤ m) are given. We prove that if ℱ is an intersecting family containing members F such that |F ∩ X1| = ki, |F ∩ X2| = ℓi holds for one of the values i (1 ≤ i ≤ m) then |ℱ| cannot exceed the size of the largest subfamily containing one element. © Wydawnictwa AGH, 2017

Strong Qualitative Independence

AbstractThe subsets A,B of the n-element X are said to be s-strongly separating if the two sets divide X into four sets of size at least s. The maximum number h(n,s) of pairwise s-strongly separating subsets was asymptotically determined by Frankl (Ars Combin. 1 (1976) 53) for fixed s and large n. A new proof is given. Also, estimates for h(n,cn) are found where c is a small constant

Around the Complete Intersection Theorem

In their celebrated paper, Erdos et al. (1961) posed the following question. Let F be a family of k-element subsets of an n-element set satisfying the condition that |F∩G|≥ℓ holds for any two members of F where ℓ≤k are fixed positive integers. What is the maximum size |F| of such a family? They gave a complete solution for the case ℓ=1: the largest family is the one consisting of all k-element subsets containing a fixed element of the underlying set. (One has to suppose 2k≤n, otherwise the problem is trivial.) They also proved that the best construction for arbitrary ℓ is the family consisting of all k- element subsets containing a fixed ℓ-element subset, but only for large n's. They also gave an example showing that this statement is not true for small n's. Later Frankl gave a construction for the general case that he believed to be the best. Frankl, Wilson and Füredi made serious progress towards the proof of this conjecture, but the complete solution was not achieved until 1996 when the surprising news came: Rudolf Ahlswede and Levon Khachatrian have found the proof. They invented the expressive name: Complete Intersection Theorem. We will show some of the consequences of this deep and important theorem. © 2016 Elsevier B.V