Generalizations of Grilletʹs theorem on maximal stable sets and maximal cliques in graphs Original Research Article

Grillet established conditions on a partially ordered set under which each maximal antichain meets each maximal chain. Berge pointed out that Grilletʹs theorem can be stated in terms of graphs, made a conjecture that strengthens it, and asked a related question. We exhibit a counterexample to the conjecture and answer the question; then we prove four theorems that generalize Grilletʹs theorem in the spirit of Bergeʹs proposals.