A property of colored complexes and their duals

Abstract 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.