**MSC:**- 06A07 Combinatorics of partially ordered sets

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 i-th level, and the shadow

of C(m,i) consists of the smallest elements of the (i-1)-st level.

P is called shadow-increasing if for all m,i the shadow 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.