Papers › Local Configurations in Union-Closed Families
Local Configurations in Union-Closed Families
Jonad Pulaj, Kenan Wood
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The Frankl or Union-Closed Sets conjecture states that for any finite union-closed family of sets ℱ containing some nonempty set, there is some element i in the ground set U(ℱ) := ⋃_(S ∈ℱ) S of ℱ such that i is in at least half of the sets in ℱ. In this work, we find new values and bounds for the least integer FC(k, n) such that any union-closed family containing FC(k, n) distinct k-sets of an n-set X satisfies Frankl's conjecture with an element of X. Additionally, we answer an older question of Vaughan regarding symmetry in union-closed families and we give a proof of a recent question posed by Ellis, Ivan and Leader. Finally, we introduce novel local configuration criteria through a generalization of Poonen's Theorem to prove the conjecture for many, previously unknown classes of families.
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