I was wondering if someone could give me a specific example of union closed family with a three element set, none of whose elements are in half the members of the family.
Thanks in advance!
I was wondering if someone could give me a specific example of union closed family with a three element set, none of whose elements are in half the members of the family. Thanks in advance! 


This type of example was first provided by Sarvate and Renaud, but I can't access that paper. The paper 'The graph formulation of the unionclosed conjecture' by Bruhn, Charbit, Schaudt, Arne Telle contains a very similar example for their graph reformulation, which is presumably converted from the SarvateRenaud example. Reconverting into the set language, this is: The family generated by the empty set, 1, 2, 3, 125, 136, 237, 567. This is a family containing 25 sets. The elements 1, 2 and 3 are each in 16 sets. The elements 5, 6 and 7 are each in 12 sets. Thus the set 567 is a three element set, none of whose elements are in at least half of the sets. (Note in case people are curious about the motivation  there is an easy argument to show that if the family contains a set with two elements then at least one of those elements is in at least half the sets. This shows the same is not true for sets of three elements.) 


This example with minimal size $3$ is from D. G. Sarvate and JC. Renaud, "Improved bounds for the unionclosed sets conjecture", Ars Combinatoria 29 (1990), pp. 181185: $\mathcal A=\{A_1,\dots,A_{27}\},$ with $A_1=\{1,2,3\}$ Here $A_1=\{1,2,3\}$ is the unique set of minimal size, and each of the elements $1,2,3$ is in exactly $13$ of the $27$ sets. 

