On the size of maximal antichains and the number of pairwise disjoint maximal chains |
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Authors: | David M Howard |
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Institution: | School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA |
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Abstract: | 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 two-dimensional which suggests that the dual statement may also hold. In this paper, we show that this is correct. Specifically, we show that if P is a finite poset and n≤|A|≤n+(n−k)/(k−2) for every maximal antichain in P, then P has k pairwise disjoint maximal chains. Our argument actually proves a somewhat stronger result, and we are able to show that an analogous result holds for antichains. |
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Keywords: | Partially ordered set Chains Antichains |
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