A property of colored complexes and their duals Original Research Article

A ranked poset P is a Macaulay poset if there is a linear order ≺ of the elements of P such that for any m, i the set C(m,i) of the m (with respect to ≺) smallest elements of rank i has minimum-sized shadow among all m-element subsets of the ith level, and the shadow of C(m,i) consists of the smallest elements of the (i−1)th level. P is called shadow-increasing if for all m, i the shadow of C(m,i) is not smaller than the shadow of C(m,i−1). We show that colored complexes and their duals, the star posets, are shadow-increasing.