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The Width of Hamming Balls

Kada Williams

Abstract

The width of a poset is the size of its largest antichain. Sperner's theorem states that $(2^{[n]},\subset)$ is a poset whose width equals the size of its largest layer. We show that Hamming ball posets also have this property. This extends earlier work that proves this in the case of small radii. Our proof is inspired by (and corrects) a result of Harper.

The Width of Hamming Balls

Abstract

The width of a poset is the size of its largest antichain. Sperner's theorem states that is a poset whose width equals the size of its largest layer. We show that Hamming ball posets also have this property. This extends earlier work that proves this in the case of small radii. Our proof is inspired by (and corrects) a result of Harper.

Paper Structure

This paper contains 3 sections, 5 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. Harper's theorem on product posets
  3. Hamming balls are LYM posets

Key Result

Theorem 2

In the poset $P=(2^{[n]},\subset)$, $w(P)=\ell(P)$.

Theorems & Definitions (15)

  • Definition 1
  • Theorem 2: Sperner, 1928
  • Definition 3
  • Definition 4
  • Theorem 5: Yamamoto-Meshalkin-Bollobás-Lubell
  • proof
  • Proposition 6: Kleitman, 1974
  • Definition 7
  • Definition 8
  • Theorem 9: Harper, 1974
  • ...and 5 more