An interpolation theorem for partitions which are complete with respect to hereditary properties

Let S be a finite set and P be a property. Then a partition {S1, …, St} of S is a complete P -partition of order t if each Si has property P but no Si υ Si (i ≠ j) has this property. P is hereditary if each subset of a set with property P has property P. The main result is an interpolation theorem for complete P-partitions where P is hereditary, viz., if S has complete P-partitions of orders m and M where m ⩽ M, then S has a complete P-partition of order n for each n, m ⩽ n ⩽ M. This result generalizes the homomorphism interpolation theorem of Harary, Heditniemi and Prins, and its proof supplies an algorithm for the construction of the interpolating partitions. There are a variety of applications to the partition theory of graphs and set systems.