A flat antichain is a collection of incomparable subsets of a finite ground set, such that |B| − |C| ≤ 1 for every two members B, C. Using Lieby's results, we prove the Flat Antichain Conjecture, which says that for any antichain there exists a flat antichain having the same cardinality and average set size. |A| A∈A |A|. We say that A is flat if for all A ∈ A we have |A| = d or |A| = d+1 for some non-negative integer d. A family B is a flat counterpart of A if B is flat, |A| = |B| and av(A)= av(B). A completely separating system (CSS) is a family C of subsets of [m] such that for each ordered pair (a, b), a = b, there is a set in C which contain a but does not contain b. It is a natural question to determine the minimum size of a k-uniform CSS: Ramsay, Roberts and Ruskey [9,10] have found upper and lower bounds on it.The dual of a set systemIt is easy to see that A is an antichain if and only if A * is a CSS. Since vol(A) = vol(A * ), a necessary condition for C ⊆ 2 [m] being a k-uniform CSS of size n is that there exists Mathematics Subject Classification (2000): 05D05
66ÁKOS KISVÖLCSEYan antichian of size m with volume kn, i.e. with average set size kn/m. Investigating this problem, Lieby [6] conjectured the following.
Flat Antichain Conjecture. If A is an antichain, then there exists a flat antichain with the same size and average set size.Thus this conjecture would make easier to check whether a CSS exists with given parameters. However, this is also a nice problem itself.The conjecture has been verified in several special cases. A result of Kleitman and Milner [5] implies that if A is an antichain with integral average set size, then the conjecture is true. Roberts [11] has solved the FAC for antichains with average set size at most 3. In her PhD thesis, Lieby has proven the conjecture if A is contained in 3 or 4 consecutive levels.
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