An interpolation theorem for partitions which are indivisible with respect to cohereditary properties (Q1179468)

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 presented is an interpolation theorem for indivisible \(P\)- partitions where \(P\) is cohereditary; that is, if \(S\) has indivisible \(P\)-partitions of orders \(n\) and \(m\), where \(n<m\), then \(S\) has an indivisible \(P\)-partition of order \(k\) for each \(k\), \(n\leq k\leq m\). The result solves a problem of \textit{E. J. Cockayne} [Theor. Appl. Graphs, Proc. Kalamazoo 1976, Lect. Notes Math. 642, 141-147 (1978; Zbl 0384.05052)].