An interpolation theorem for partitions which are indivisible with respect to cohereditary properties

Abstract

Let S be a finite set and P be a property associated with the subsets of S. Then a partition { S 1, …, S k } of S is an indivisible P-partition of order k if each S i has property P but no S i is a union of two disjoint sets with property P. P is cohereditary if each superset of a set with property P has property P. The main result is an interpolation theorem for indivisible P-partitions where P is cohereditary, viz., if S has indivisible P-partitions of order n and m, where n < m, then S has an indivisible P-partition of order k for each k, n ≤ k ≤ m. This result is an analogy of the interpolation theorem for complete P-partitions of Cockayne, Miller, and Prins. The result solves one problem of Cockayne.