Progress on the union-closed conjecture and offsprings in winter 2022-2023

TL;DR

This paper discusses recent progress on the long-standing union-closed conjecture, highlighting a breakthrough that established a lower bound on the frequency of the most common element in such families.

Contribution

It reports a significant breakthrough by Justin Gilmer that provides the first constant lower bound for the proportion of the most common element in union-closed families.

Findings

Established a constant lower bound for element frequency

Increased understanding of union-closed conjecture variants

Triggered renewed research interest in the conjecture

Abstract

Mathematicians had little idea whether the easy-to-state union-closed conjecture was true or false even after $40$ years. However, last winter saw a surge of interest in the conjecture and its variants, initiated by the contribution of a researcher at Google. Justin Gilmer [arXiv:2211.09055] made a significant breakthrough by discovering a first constant lower bound for the proportion of the most common element in a union-closed family.