Bounding the consistency strength of a five element linear basis

In 13] it was demonstrated that the Proper Forcing Axiom implies that there is a five element basis for the class of uncountable linear orders. The assumptions needed in the proof have consistency strength of at least infinitely many Woodin cardinals. In this paper we reduce the upper bound on the consistency strength of such a basis to something less than a Mahlo cardinal, a hypothesis which can hold in the constructible universe L. A crucial notion in the proof is the saturation of an Aronszajn tree, a statement which may be of broader interest. We show that if all Aronszajn trees are saturated and PFA(ω ₁) holds, then there is a five element basis for the uncountable linear orders. We show that PFA(ω ₂) implies that all Aronszajn trees are saturated and that it is consistent to have PFA(ω ₁) plus every Aronszajn tree is saturated relative to the consistency of a reflecting Mahlo cardinal. Finally we show that a hypothesis weaker than the existence of a Mahlo cardinal is sufficient to force the existence of a five element basis for the uncountable linear orders. The first author acknowledges a fellowship granted by the French ministry of research. The research of the second author was partially supported by the Centre de Rercerca Matemàtica of the Universitat Autònoma de Barcelona, and by NSF Grant DMS-0401603. The second author would also like to thank the third and fourth authors for bringing him to Boise and Paris respectively for further discussions. The third author was supported by NSF grants DMS-0401893 and DMS-0200671. The second and fourth authors would like to thank CIRM in Luminy for hosting them during a petit group de travaille, and to thank the others participants, Ralf Schindler and Ernest Schimmerling, for discussions on this topic.