Let ℱ be the class of complete, finitely axiomatizable ω-categorical theories. It is not known whether there are simple theories in ℱ. We prove three results of the form: if T∈ ℱ has a sufficently well-behaved definable set J, then T is not simple. (In one case, we actually prove that T has the strict order property.) All of our arguments assume that the definable set J satisfies the Mazoyer hypothesis, which controls how an element of J can be algebraic over a subset of the model. For every known example in ℱ, there is a definable set satisfying the Mazoyer hypothesis.
"Finitely axiomatizable ω-categorical theories and the Mazoyer hypothesis." J. Symbolic Logic 70 (2) 460 - 472, June 2005. https://doi.org/10.2178/jsl/1120224723