Generalized Pigeonhole Properties of Graphs and Oriented Graphs

A relational structure A satisfies the P(n, k) property if whenever the vertex set of A is partitioned into n nonempty parts, the substructure induced by the union of somek of the parts is isomorphic to A. The P(2, 1) property is just the pigeonhole property, (P), introduced by Cameron, and studied by Bonato, Delić and Cameron. We classify the countable graphs, tournaments, and oriented graphs with the P(3, 2) property.