Further analysis on the second frequency of union-closed set families

TL;DR

This paper investigates conditions under which Nagel's strengthened union-closed sets conjecture fails for the case k=2, using linear programming methods to analyze specific family structures.

Contribution

It provides a detailed analysis of when Nagel's conjecture does not hold for k=2, extending previous results with a linear programming approach.

Findings

Identifies specific cases where Nagel's conjecture fails for k=2

Uses linear programming to analyze union-closed set families

Extends understanding of the conjecture's limitations

Abstract

The Union-Closed Sets Conjecture, also known as Frankl's conjecture, asks whether, for any union-closed set family $\mathcal{F}$ with $m$ sets, there is an element that lies in at least $\frac{1}{2}\cdot m$ sets in $\mathcal{F}$. In 2022, Nagel posed a stronger conjecture that within any union-closed family whose ground set size is at least $k$, there are always $k$ elements in the ground set that appear in at least $\frac{1}{2^{k-1}+1}$ proportion of the sets in the family. Das and Wu showed that this conjecture is true for $k\geq 3$ and $k=2$ if $|\mathcal{F}|$ is outside a particular range. In this companion paper, we analyse further when $\mathcal{F}$ fails Nagel's conjecture for $k=2$ via linear programming.