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

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 $k\ge 3$ and $k=2$ if $\mathrm{\mid }\mathcal{F}\mathrm{\mid }$ 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.