A sequential coloring algorithm for finite sets

Let X be a finite set and P a hereditary property associated with the subsets of X. A partition of X into n subsets each with property P is said to be a P-n-coloring of X. The minimum n such that a P-n-coloring of X exists is defined to be the P-chromatic number of X. In this paper we give a sequential coloring algorithm for P-coloring X. From the algorithm we then get a few upper bounds for the P-chromatic number. In particular, we generalize the Welsh-Powell upper bound for ordinary chromatic number to the case of P-chromatic number of any finite set X.