Papers › Further analysis on the second frequency of union-closed set families
Further analysis on the second frequency of union-closed set families
Saintan Wu
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The Union-Closed Sets Conjecture, also known as Frankl's conjecture, asks whether, for any union-closed set family ℱ with m sets, there is an element that lies in at least 1/2·m sets in 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 1/(2ᵏ⁻¹+1) proportion of the sets in the family. Das and Wu showed that this conjecture is true for k≥3 and k=2 if |ℱ| is outside a particular range. In this companion paper, we analyse further when ℱ fails Nagel's conjecture for k=2 via linear programming.
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