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.