On the simplicity of ordered geometry (Q1956325)

In the axiomatic framework of ordered planes, all axioms can be expressed with at most four variables, with the exception of the Pasch axiom, which requires six variables. The author shows that the Pasch axiom can be replaced by the conjunction of Pasch's theorem with the weak crossbar theorem, which require six resp. five variables for their formulation. This yields a simplest axiom system for ordered planes. The question whether it is possible to replace the Pasch axiom by axioms which require only five variables remains open.