Abstract
A family ℱ of sets has propertyB if there exists a setS such thatS∩F≠0 andS⊄F for everyF∈ℱ. ℱ has propertyB(s) if there exists a setS such that 0<|F∩S|<s for everyF∈ℱ. Denote bym(n) (respectivelym(n, s)) the size of a smallest family ofn-element sets not having propertyB (respectivelyB(s)). P. Erdős has asked whetherm(n, s)≧m (s) for alln≧s. We show that, in general, this inequality does not hold.
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Abbott, H.L., Liu, A. On a problem of Erdős concerning property B. Combinatorica 7, 215–219 (1987). https://doi.org/10.1007/BF02579298
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DOI: https://doi.org/10.1007/BF02579298