منابع مشابه
Maximal Chains and Antichains in Boolean Lattices
The following equivalent results in the Boolean lattice 2 are proven. (a) Every fibre of 2 contains a maximal chain. (b) Every cutset of 2 contains a maximal antichain. (c) Every red-blue colouring of the vertices of 2 produces either a red maximal chain or a blue maximal antichain. (d) Given any n antichains in 2 there is a disjoint maximal antichain. Statement (a) is then improved to: (a') Ev...
متن کاملOn the size of maximal antichains and the number of pairwise disjoint maximal chains
Fix integers n and k with n ≥ k ≥ 3. Duffus and Sands proved that if P is a finite poset and n ≤ |C| ≤ n+ (n− k)/(k− 2) for every maximal chain in P , then P must contain k pairwise disjoint maximal antichains. They also constructed a family of examples to show that these inequalities are tight. These examples are 2-dimensional which suggests that the dual statement may also hold. In this paper...
متن کاملMaximal Antichains of Minimum Size
Let n > 4 be a natural number, and let K be a set K ⊆ [n] := {1, 2, . . . , n}. We study the problem to find the smallest possible size of a maximal family A of subsets of [n] such that A contains only sets whose size is in K, and A 6⊆ B for all {A,B} ⊆ A, i.e. A is an antichain. We present a general construction of such antichains for sets K containing 2, but not 1. If 3 ∈ K our construction a...
متن کاملMaximal antichains under dimension constraints
We consider A(n, k) = maxA{|A| : dim(A) ≤ k, A ⊂ {0, 1} n is an antichain}, where the dimension is taken from the linear span of A in R, we conjecture the exact value of A(n, k) and we prove this conjecture for all n and k ≤ ⌊ n 2 ⌋ +1 or k = n−1. This is a contribution to the program of systematic investigation of extremal problems under dimension constraints, which was recently presented by t...
متن کاملChains and antichains in partial orderings
We study the complexity of infinite chains and antichains in computable partial orderings. We show that there is a computable partial ordering which has an infinite chain but none that is Σ1 or Π 1 1, and also obtain the analogous result for antichains. On the other hand, we show that every computable partial ordering which has an infinite chain must have an infinite chain that is the differenc...
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ژورنال
عنوان ژورنال: Fundamenta Mathematicae
سال: 1969
ISSN: 0016-2736,1730-6329
DOI: 10.4064/fm-65-2-157-167