A New Upper Bound for Cancellative Pairs

TL;DR

This paper improves the upper bound on the size of cancellative pairs of set families, reducing it from approximately 2.3264^n to 2.2682^n, and also advances bounds related to Simonyi's sandglass conjecture.

Contribution

The authors establish a tighter upper bound for the maximum size of cancellative pairs, enhancing previous bounds and contributing to related conjectures in set system theory.

Findings

Improved upper bound for cancellative pairs to 2.2682^n.

Enhanced bounds for Simonyi's sandglass conjecture.

Demonstrated limitations on the size of cancellative pairs.

Abstract

A pair $(\mathcal{A},\mathcal{B})$ of families of subsets of an $n$-element set is called cancellative if whenever $A,A'\in\mathcal{A}$ and $B\in\mathcal{B}$ satisfy $A\cup B=A'\cup B$, then $A=A'$, and whenever $A\in\mathcal{A}$ and $B,B'\in\mathcal{B}$ satisfy $A\cup B=A\cup B'$, then $B=B'$. It is known that there exist cancellative pairs with $|\mathcal{A}||\mathcal{B}|$ about $2.25^n$, whereas the best known upper bound on this quantity is $2.3264^n$. In this paper we improve this upper bound to $2.2682^n$. Our result also improves the best known upper bound for Simonyi's sandglass conjecture for set systems.