# The sandglass conjecture beyond cancellative pairs

## TL;DR

This paper advances the understanding of the sandglass conjecture by establishing a new upper bound on the product of sizes of recovering families, surpassing previous bounds and addressing a key theoretical barrier.

## Contribution

It introduces an exponential separation between bounds on recovering pairs and cancellative pairs, leading to a tighter upper bound on the sandglass conjecture.

## Key findings

- New upper bound: |A| |B| ≤ 2.2543^n
- Exponential separation between recovering and cancellative pairs
- Improved understanding of the sandglass conjecture's limits

## Abstract

The sandglass conjecture, posed by Simonyi, states that if a pair $(A, B)$ of families of subsets of $[n]$ is recovering then $|A| |B| \leq 2^n$. We improve the best known upper bound to $|A| |B| \leq 2.2543^n$. To do this we overcome a significant barrier by exponentially separating the upper bounds on recovering pairs from cancellative pairs, a related notion.