Two-colorings with many monochromatic cliques in both colors

Color the edges of the n-vertex complete graph in red and blue, and suppose that red k-cliques are fewer than blue k-cliques. We show that the number of red k-cliques is always less than cknk, where ck∈(0, 1) is the unique root of the equation zk=(1-z)k+kz(1-z)k-1. On the other hand, we construct a coloring in which there are at least cknk-O(nk-1) red k-cliques and at least the same number of blue k-cliques. © 2013 Elsevier Inc

Linear independence, a unifying approach to shadow theorems

The intersection shadow theorem of Katona is an important tool in extremal set theory. The original proof is purely combinatorial. The aim of the present paper is to show how it is using linear independence latently. © 2016 Elsevier Ltd

New inequalities for families without k pairwise disjoint members

Some best possible inequalities are established for k-partition free families (cf. Definition 1) and they are applied to prove a sharpening of a classical result of Kleitman concerning families without k pairwise disjoint members. (C) 2018 Elsevier Inc. All rights reserved

Shattered matchings in intersecting hypergraphs

Let X be an n-element set, where n is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family F of n2-element subsets of X, one can partition X into n2 disjoint pairs in such a way that no matter how we pick one element from each of the first n2−1 pairs, the set formed by them can always be completed to a member of F by adding an element of the last pair. The above problem is related to classical questions in extremal set theory. For any t≥2, we call a family of sets F⊂2X {\em t-separable} if for any ordered pair of elements (x,y) of X, there exists F∈F such that F∩{x,y}={x}. For a fixed t,2≤t≤5 and n→∞, we establish asymptotically tight estimates for the smallest integer s=s(n,t) such that every family F with |F|≥s is t-separable