Large Bd-free and union-free subfamilies

János Barát, Zoltán Füredi, Ida Kantor, Younjin Kim, Balázs Patkós

Research output: Article


For a property Γ and a family of sets F, let f(F,Γ) be the size of the largest subfamily of F having property Γ. For a positive integer m, let f(m, Γ) be the minimum of f(F,Γ) over all families of size m. A family F is said to be Bd-free if it has no subfamily F'={FI:I⊆[d]} of 2d distinct sets such that for every I, J⊆[d], both FI∪FJ=FI∪J and FI∩FJ=FI∩J hold. A family F is a-union free if F1∪⋯∪Fa≠Fa+1 whenever F1,...,Fa+1 are distinct sets in F. We verify a conjecture of Erdos and Shelah that f(m, B2-free)=Θ(m2/3). We also obtain lower and upper bounds for f(m, Bd-free) and f(m, a-union free).

Original languageEnglish
Pages (from-to)101-104
Number of pages4
JournalElectronic Notes in Discrete Mathematics
Publication statusPublished - dec. 1 2011


ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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