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The Width of a Ball in a Hypercube

Kada Williams

Abstract

Various authors have calculated how many pairwise incomparable points can be selected from a partially ordered set. We tackle this question for the family of subsets of a finite set obtained by removing or adding a bounded number of elements from a given subset. Our versatile approach is proven valid under the condition of the set size exceeding the cube of the radius.

The Width of a Ball in a Hypercube

Abstract

Various authors have calculated how many pairwise incomparable points can be selected from a partially ordered set. We tackle this question for the family of subsets of a finite set obtained by removing or adding a bounded number of elements from a given subset. Our versatile approach is proven valid under the condition of the set size exceeding the cube of the radius.
Paper Structure (3 sections, 3 theorems, 7 equations, 2 figures)

This paper contains 3 sections, 3 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Sublayers
  3. Conclusion

Key Result

Lemma 3

In the poset $B_r[p,q]$, let $\mathcal{C}$ be a weighted collection of chains that contains a set in every layer. Of those, let $\mathcal{C}_{i,j}$ contain the chains that pass through $X_{i,j}$. In view of the proportions $\frac{|\mathcal{C}_{i,j}|}{|X_{i,j}|}$, if there is a layer such that these

Figures (2)

  • Figure 1: The sublayers of $B_4[5,8]$ and its largest layer
  • Figure 2: In $B_{10}[9,17]$, sphere layers decrease downwards from below the divide

Theorems & Definitions (9)

  • Definition 1
  • Conjecture 2
  • Lemma 3
  • proof
  • proof
  • Corollary 5
  • proof
  • Theorem 6
  • proof