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Flattening antichains with respect to the volume. (English) Zbl 0913.05092
Electron. J. Comb. 6, No. 1, Research paper R1, 12 p. (1999); printed version J. Comb. 6, 1-12 (1999).
Given a (finite) poset $$(P,\leq)$$ with a strictly increasing function $$h:P\to N=\{0,1,2, \dots\}$$, define the volume of a subset $$S$$ of $$P$$ (with respect to $$h$$) to be $$\sum_{x\in S} h(x)$$. A subset $$S$$ of $$P$$ is flat (with respect to $$h)$$ if $$x,y\in S$$ with $$h(x)<h(y)$$ implies there is no $$z\in P$$ such that $$h(x)< h(z)<h(y)$$. Thus, if $$P=B_n$$, the lattice of subsets of $$[n]=\{1, \dots, n\}$$, and $$h(x)= | x|$$, then $$S$$ is flat if there exists a $$k$$ such that every element of $$S$$ has cardinality $$k$$ or $$k+1$$. There are many posets which are themselves flat with respect to some function $$h$$. Many more (e.g., chains) have flat antichains only. It has been conjectured that $$B_n$$ is among the posets which have the property that for every antichain there is a flat antichain with the same volume $$(h(x)= | x|)$$. Special cases occur in the literature. In this paper the conjecture is settled positively via a sequence of quite elegant technical lemmas based on results by Sperner, Clements and Daykin, among others. Rather than closing a area of inquiry it may also be the case that this paper can be adapted to a more general situation and lead to other interesting conjectures.

##### MSC:
 05D99 Extremal combinatorics 06A07 Combinatorics of partially ordered sets 06E99 Boolean algebras (Boolean rings)
##### Keywords:
poset; volume; flat; antichain
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