A Generalisation of Sperner's Theorem Using Weighted Chains

Yaël Dillies; Matthew Johnson; Aleksandra Kowalska

A Generalisation of Sperner's Theorem Using Weighted Chains

Combinatorics 2025-10-01 v1

Authors: Yaël Dillies , Matthew Johnson , Aleksandra Kowalska

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.

Keywords

set theory and dimension theory combinatorial bounds and extremal problems arithmetic progressions

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@article{arxiv.2509.26493,
  title  = {A Generalisation of Sperner's Theorem Using Weighted Chains},
  author = {Yaël Dillies and Matthew Johnson and Aleksandra Kowalska},
  journal= {arXiv preprint arXiv:2509.26493},
  year   = {2025}
}

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29 pages, 8 figures

A Generalisation of Sperner's Theorem Using Weighted Chains

Abstract

Keywords

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