A Property Of Colored Complexes And Their Duals

. A ranked poset P is a Macaulay poset if there is a linear order OE of the elements of P such that for any m; i the set C(m; i) of the m (with respect to OE) smallest elements of rank i has minimum--sized shadow among all m--element subsets of the i--th level, and the shadow of C(m; i) consists of the smallest elements of the (i \Gamma 1)--st 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 \Gamma 1). We show that colored complexes and their duals, the star posets, are shadow--increasing. 1. Introduction 1.1. Macaulay posets. Let P be a ranked poset with the associated partial order , and let N i (P ) be its i-th level. (For ranked poset see [9].) For any x 2 N i (P ) the shadow of x is the set \Delta x of all y 2 N i\Gamma1 (P ) such that y x, and for X ` N i (P ) the shadow \Delta X is the union of all \Delta x with x 2 X. P is said to be a Macaulay poset if there exists a linear order OE of its elements such..