union_closed.variants.univ_card

Vuckovic and Zivkovic [Vu17] showed that the union-closed sets conjecture holds for set families whose universal set has cardinality at most 12. [Vu17] Vuckovic, Bojan; Zivkovic, Miodrag (2017). "The 12-Element Case of Frankl's Conjecture" (PDF). IPSI BGD Transactions on Internet Research. 13 (1): 65.

theorem union_closed.variants.univ_card
    [Fintype n] [Nonempty n]
    (h_ne_singleton_empty : A ≠ {∅})
    (h_union_closed : IsUnionClosed A)
    (h_card : Fintype.card n ≤ 12) :
    ∃ i : n, (1 / 2 : ℚ) * #A ≤ #{x ∈ A | i ∈ x}

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