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A Generalisation of Sperner's Theorem Using Weighted Chains

Yaël Dillies, Matthew Johnson, Aleksandra Kowalska

TL;DR

This work generalizes Sperner-type extremal set theory to the cube {0,1,...,d}^n under a width constraint k. It develops a weighted symmetric chain decomposition for the d=1 and d=2 cases, proving that the unique (for d=2) largest family is the layer set {|x| ≡ n (mod 2k+1)}. The method leverages explicit chain weights to ensure a unit induced weight on all points, with positivity proved via recursive binomial-type identities; this yields a constructive optimal set and a new Sperner proof as a byproduct. The authors also outline conjectures and asymptotic results suggesting similar parity-based optimal sets for general d and discuss potential generalizations and future research directions.

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.

A Generalisation of Sperner's Theorem Using Weighted Chains

TL;DR

This work generalizes Sperner-type extremal set theory to the cube {0,1,...,d}^n under a width constraint k. It develops a weighted symmetric chain decomposition for the d=1 and d=2 cases, proving that the unique (for d=2) largest family is the layer set {|x| ≡ n (mod 2k+1)}. The method leverages explicit chain weights to ensure a unit induced weight on all points, with positivity proved via recursive binomial-type identities; this yields a constructive optimal set and a new Sperner proof as a byproduct. The authors also outline conjectures and asymptotic results suggesting similar parity-based optimal sets for general d and discuss potential generalizations and future research directions.

Abstract

We find the (unique) largest subset of such that it contains no two elements, one of which is coordinatewise greater than the other, but strictly greater on at most 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.

Paper Structure

This paper contains 26 sections, 19 theorems, 90 equations, 8 figures.

Table of Contents

  1. The problem statement and initial motivation
  2. Acknowledgements
  3. Background and related research
  4. Motivation for the problem
  5. A similar problem in a different metric
  6. Why the methods used to solve Problem \ref{['the_main_problem']} are different from those used for Problem \ref{['another_problem']}.
  7. The d=1 case
  8. Definitions and notation
  9. Weighted chains
  10. Assigning chain weights so that the induced weights are all 1
  11. Weights of chains starting in outer layers are positive
  12. Weights of chains starting in inner layers are positive
  13. The d=2 case
  14. Definitions and notation
  15. Plotting the chains in the cube
  16. ...and 11 more sections

Key Result

Theorem 1

For any $0 \leqslant k \leqslant n$, the unique largest subset $A$ of $\{0, 1, 2\}^n$ s.t. there are no two distinct elements $x, y \in A$ with $x \leqslant y$ coordinatewise and $x_i < y_i$ on at most $k$ coordinates, is where $|x|=\sum_i x_i$.

Figures (8)

  • Figure 1: $n=10, k=3$
  • Figure 2: $n=13, k=5$
  • Figure 3: $n=9$
  • Figure 4: $n=9, k=2$
  • Figure 5: $n=13, k=2$. Contributions for $W(6, 3)$ according to Equation \ref{['key_observation']}.
  • ...and 3 more figures

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 7
  • proof
  • Definition 8
  • Definition 9
  • ...and 33 more