On disjoint chains of subsets

摘要

We prove the following theorem concerning the poset of all subsets of [n] ordered by inclusion. Consider any two equal-size families of subsets of [n], J and R, where within each Family all subsets have the same number of elements. Suppose there exists a bijection phi: J --> R such that A superset of F(A) for all A is an element of J. Then there exist J disjoint saturated chains containing all the subsets in J and R. (C) 2001 Academic Press. [References: 4]