Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sizes of flat maximal antichains of subsets

Jerrold R. Griggs, Thomas Kalinowski, Uwe Leck, Ian T. Roberts, Michael Schmitz

TL;DR

The paper addresses the problem of characterizing which sizes $m$ occur for maximal antichains in the Boolean lattice $B_n$, focusing on flat maximal antichains spanning two consecutive levels. It develops a framework based on shadow concepts, squashed order, and Kruskal-Katona, and proves that for a wide range of $m$, there exist flat maximal antichains of size $m$ on levels $l$ and $l+1$ with $2\le l\le (n-2)/2$. By establishing recurrences and constructing expansive intervals across levels, the authors show that almost all sizes in the feasible range arise from flat two-level constructions, thereby enriching the understanding of the spectrum $S(n)$. The results connect combinatorial constructions with graph-theoretic and coding-theoretic tools to provide a near-complete description of maximal antichain sizes and suggest directions for exact determinations and further refinements.

Abstract

This is the second of two papers investigating for which positive integers $m$ there exists a maximal antichain of size $m$ in the Boolean lattice $B_n$ (the power set of $[n]:=\{1,2,\dots,n\}$, ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only $l$-sets and $(l+1)$-sets for some $l$.

Sizes of flat maximal antichains of subsets

TL;DR

The paper addresses the problem of characterizing which sizes occur for maximal antichains in the Boolean lattice , focusing on flat maximal antichains spanning two consecutive levels. It develops a framework based on shadow concepts, squashed order, and Kruskal-Katona, and proves that for a wide range of , there exist flat maximal antichains of size on levels and with . By establishing recurrences and constructing expansive intervals across levels, the authors show that almost all sizes in the feasible range arise from flat two-level constructions, thereby enriching the understanding of the spectrum . The results connect combinatorial constructions with graph-theoretic and coding-theoretic tools to provide a near-complete description of maximal antichain sizes and suggest directions for exact determinations and further refinements.

Abstract

This is the second of two papers investigating for which positive integers there exists a maximal antichain of size in the Boolean lattice (the power set of , ordered by inclusion). In the first part, the sizes of maximal antichains have been characterized. Here we provide an alternative construction with the benefit of showing that almost all sizes of maximal antichains can be obtained using antichains containing only -sets and -sets for some .
Paper Structure (10 sections, 18 theorems, 72 equations, 9 figures, 1 table)

This paper contains 10 sections, 18 theorems, 72 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction
  2. High level overview
  3. Large maximal flat antichains
  4. Recurrences
  5. The base case
  6. Intervals in
  7. Proofs of and
  8. Proof of Proposition \ref{['prop:flat']}
  9. Proof of
  10. Open problems

Key Result

Theorem 1

Let $k=\lceil n/2\rceil$, and let $m$ be an integer with $1\leqslant m\leqslant\binom{n}{k}$. Then $m\in S(n)$ if and only if $m\leqslant\binom{n}{k}-k\left\lceil(k+1)/2\right\rceil$ or $m$ has the form for some integers $l\in\{k,n-k\}$, $t\in\{0,\dots,k\}$, $a\geqslant b\geqslant c\geqslant 0$, $a+b\leqslant t$. Moreover, if $m\in S(n)$ satisfies $m\geqslant\binom{n}{k}-k^2$ then there exists a

Figures (9)

  • Figure 1: The edges of the graphs are the complements of the members of $\mathcal{F}_0$, $\mathcal{F}_1$, $\mathcal{F}_2$.
  • Figure 2: The edges of the graphs are the complements of the members of $\mathcal{F}_3$, $\mathcal{F}_4$, $\mathcal{F}_5$.
  • Figure 3: The edges of the graphs are the complements of the members of $\mathcal{F}_6$, $\mathcal{F}_7$, $\mathcal{F}_8$.
  • Figure 4: The edges of the graphs are the complements of the members of $\mathcal{F}_9$ and $\mathcal{F}_{10}$.
  • Figure 5: The dots correspond to the squashed maximal flat antichains $\mathcal{A}\subseteq\binom{[13]}{10}\cup\binom{[13]}{11}$. In the left plot, the segments correspond to the maximal antichains $\mathcal{A}_{i,t}$ ($i=0,1,\dots,10$, $t=0,1,\dots,10$). In the right plot, we have also added the maximal flat antichains obtained by recursion from $n=12$. The interval $[73,237]$ of sizes of maximal antichains contained in $\binom{[13]}{11}\cup\binom{[13]}{10}$ is indicated by horizontal lines.
  • ...and 4 more figures

Theorems & Definitions (37)

  • Theorem 1: Griggs2023
  • Theorem 2
  • Corollary 1
  • Proposition 1
  • Proposition 2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • ...and 27 more