union_closed.variants.family_card

Roberts and Simpson [Ro10] showed that the union-closed sets conjecture holds for set families of size at most 46. Their method, however, combined with the result of [Vu17], further shows that it holds for `#A ≤ 50` as well. [Ro10] Roberts, Ian; Simpson, Jamie (2010). "A note on the union-closed sets conjecture" (PDF). Australas. J. Combin. 47: 265–267. [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.family_card
    [Nonempty n]
    (h_ne_singleton_empty : A ≠ {∅})
    (h_union_closed : IsUnionClosed A)
    (hA : #A ≤ 50) :
    ∃ i : n, (1 / 2 : ℚ) * #A ≤ #{x ∈ A | i ∈ x}

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