New simple theories from hypergraph sequences

We develop a family of simple rank one theories built over quite arbitrary sequences of finite hypergraphs. (This extends an idea from the recent proof that Keisler’s order has continuum many classes, however, the construction does not require familiarity with the earlier proof.) We prove a model-completion and quantifier-elimination result for theories in this family and develop a combinatorial property which they share. We invoke regular ultrafilters to show the strength of this property, showing that any flexible ultrafilter which is good for the random graph is able to saturate such theories.