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Set theory question!

  1. Sep 25, 2012 #1
    1. The problem statement, all variables and given/known data
    Hi everyone, i have the following problem, that for the moment i couldn't find the solution or even how to search about...

    This is it, we have a family of subsets F ={F_1,...,F_p} of the set {1,2,...,k}.

    We know that, for every i != j, F_i !=F_j
    and for every pair of elements a, b \in {1,2,...k} there exists one F_i
    such that a belongs to F_i but b does not.
    (this is, no pair of elements belong to exactly the same sets)

    Notice that with this restriction, at most one element can be outside every F_i.

    it is clear that p should be at least ceiling ( log_2 ( k ) ).

    First question. Does this family of sets have any name in the literature?

    Second question:
    Consider now the maximal sets S_1,S_2,...S_s such that
    S_i is a subset of F and the intersection of every F_j \in S_i is non-empty.
    The maximality means that there is no S_i subset of S_j with exactly
    the same intersection, i.e., the intersection of F_x \in S_i != intersection of F_x \in S_j
    for S_i subset of S_j.

    I want to prove that s >= k-1

    Thank you all for any answer :)
    Juan!


    2. Relevant equations



    3. The attempt at a solution
    No idea how to solve but just in particular cases, for example for p = k-1
    and F_i ={i}....
     
  2. jcsd
  3. Sep 26, 2012 #2
    mmh nothing...? :(
     
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