About the ratio of the size of a maximum antichain to the size of a maximum level in finite partially ordered sets

Abstract

LetP be a finite partially ordered set. The lengthl(x) of an elementx ofP is defined by the maximal number of elements, which lie in a chain withx at the top, reduced by one. Letw(P) (d(P)) be the maximal number of elements ofP which have the same length (which form an antichain). Further let\(p^n : = \underbrace {PX...XP}_{n - times}\). The numbers\(r_k : = \mathop {\max }\limits_{P:|P| = k} \frac{{d(P)}}{{w(P)}}\) and\(s_k : = \mathop {\max }\limits_{P:|P| = k} \mathop {\lim }\limits_{n \to \infty } \frac{{d(P^n )}}{{w(P^n )}}\) as well as all partially ordered sets for which these maxima are attained are determined.

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Engel, K., Kuzjurin, N.N. About the ratio of the size of a maximum antichain to the size of a maximum level in finite partially ordered sets. Combinatorica 5, 301–309 (1985). https://doi.org/10.1007/BF02579244

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AMS subject classification (1980)

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  • 05 A 99