We call a system of non-empty sets H entwined, if for every disjoint pair of sets A and B in H there exists b∈B such that A∪{b} is in H or there exists a∈A such that B∪{a} is in H.Let H be an entwined system of sets containing all of {1},{2},…,{n}. Prove that if n>k(k+1)/2, then H contains a set with at least k+1 elements, and this is sharp for every k, i.e. if n=k(k+1), it is possible that every set in H has at most k elements. komalcombinatoricsset theory