2010/12/17 by János Barát, Zoltán Füredi, Barát, János +7
Mathematics · #05D05 #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1012.3918
openalex publication_date 2010/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a property Γ and a family of sets \cF, let f(\cF,Γ) be the size of the largest subfamily of \cF having property Γ. For a positive integer m, let f(m,Γ) be the minimum of f(\cF,Γ) over all families of size m. A family \cF is said to be Bd-free if it has no subfamily \cF'=\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 \cF is a-union free if F1∪ ... Fa ≠ Fa+1 whenever F1,..,Fa+1 are distinct sets in \FF. We verify a conjecture of Erd\H os and Shelah that f(m, B2\rm -free)=Θ(m2/3). We also obtain lower and upper bounds for f(m, Bd\rm -free) and f(m,a\rm -union free).