A generalization of the independence number

Jianguo Qian, Konrad Engel and Wei Xu (Dass et al., 2015) gave a generalization of Sperner's theorem (Sperner, 1928): n and m are given integers, they found the minimum number of pairs Y-i subset of Y-j(i not equal j) in a multifamily {Y-1, ..., Y-m}of not necessarily different subsets of an n-element set. Here a far reaching generalization and easier proof is given. Let G be a graph and m an integer, choose m vertices with possible repetitions in such a way that the number of adjacent pairs (including the repeated vertices) is minimum. It is proved that the following choice gives the minimum: take the vertices of a largest independent set in G with nearly equal multiplicities. (C) 2022 The Author(s). Published by Elsevier B.V.