On an acyclic relaxation of incomparable families of sets

TL;DR

This paper investigates the structure and limits of sequences of set families with specific incomparability properties, establishing bounds on their sizes and lengths, and introduces an acyclic relaxation approach for these families.

Contribution

It introduces an acyclic relaxation framework for incomparable families of sets and determines bounds on the maximum size and length of such sequences.

Findings

Maximum size of two incomparable families is 1/4 of 2^k.

Long d-exceeding sequences can be constructed with size close to 1/2 of 2^k.

Bounds are established for the maximum length of d-exceeding sequences.

Abstract

For two families $\mathcal{A}, \mathcal{B} \subseteq \mathcal{P}([k])$, we write $\mathcal{A}\vdash\mathcal{B}$ if $A\not\supseteq B$ for each two sets $A \in \mathcal{A}$ and $B \in \mathcal{B}$. $\mathcal{A}$ and $\mathcal{B}$ are called incomparable if $\mathcal{A}\vdash\mathcal{B}$ and $\mathcal{B}\vdash\mathcal{A}$. Seymour proved that the maximum size of two incomparable equal-sized families in $\mathcal{P}([k])$ is $\frac{1}{4}2^k$. A sequence of families $\mathcal{B}_1,\dots,\mathcal{B}_l \ \subseteq \mathcal{P}([k])$ is called $d$-exceeding if $\mathcal{B}_i\vdash\mathcal{B}_j$ for all $i,j\in [l]$ with $j-i\in [d]$. Cyclically reusing $d+1$ pairwise incomparable families yields arbitrarily long $d$-exceeding sequences of families. We prove inversely that the maximum size of equal-sized families of a sufficiently long $1$-exceeding sequence in $\mathcal{P}([k])$ is also $\frac{1}{4}2^k$. A sequence of sets $B_1,\dots,B_l \subseteq [k]$ is called $d$-exceeding if $\{B_1\},\dots,\{B_l\}$ is $d$-exceeding, that is, if $B_i \not\supseteq B_j$ for all $i,j\in [l]$ with $j-i\in [d]$. We locate the maximum $d$ such that there exist arbitrarily long $d$-exceeding sequences of subsets of $[k]$ between $(1-o(1)) \tfrac{1}{e} 2^k$ and $\tfrac{1}{2}2^k-2$.