A Generalisation of Sperner's Theorem Using Weighted Chains

Abstract

We find the (unique) largest subset of $\{0, 1, 2\}^n$ such that it contains no two elements, one of which is coordinatewise greater than the other, but strictly greater on at most $k$ coordinates. To do so, we decompose the cube into weighted chains. In Appendix B we present a new proof of Sperner's theorem we found while working on this problem.