Almost $k$-union closed set systems

TL;DR

This paper extends the concept of almost union-closed set systems to higher order unions, building on Gilmer's recent breakthrough and Chase and Lovett's optimal results, advancing understanding in combinatorial set systems.

Contribution

The paper generalizes the almost union-closed set system results to higher order unions, expanding the scope of previous foundational work.

Findings

Extended the almost union-closed set system results to higher order unions.

Built upon Gilmer's and Chase-Lovett's methods to achieve this generalization.

Contributed to the theoretical understanding of union-closed set systems.

Abstract

In a recent breakthrough, Gilmer proved the union closed conjecture up to a constant factor. Using Gilmer's method and additional ideas, Chase and Lovett proved an optimal result for almost union-closed set systems. Here that result is extended to higher order unions.