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'''Theorem (Greene-Kleitman<ref>C. Greene, D. J. Kleitman, ''The structure of Sperner k-families'', J. Combinatorial Theory, Ser.A 20 (1976), 41-68.<name="GrKl"/ref>).''' ''Let <math>(V,\leq)</math> be a partially ordered finite set, and let k be a positive integer. Then the maximum size of the union of k antichains equals the minimum of <math>\sum_{C \in \Pi} \min\{|C|,k\}</math> taken over all partitions <math>\Pi</math> of V into chains.''
An immediate corollary is the following: if ''X'' is a maximum size union of ''k'' antichains, and <math>\Pi</math> is a chain partition that minimizes <math>\sum_{C \in \Pi} \min\{|C|,k\},</math> then every chain <math>C \in \Pi</math> contains exactly <math>\min\{|C|,k\}</math> elements from ''X''.
==References==
<references><ref name="GrKl">C. Greene, D. J. Kleitman, ''The structure of Sperner k-families'', J. Combinatorial Theory, Ser. A 20 (1976), 41-68, [http://dx.doi.org/10.1016/0097-3165(76)90077-7 DOI link]</ref></references>
[[Category:Theorems]]
