Graph models are simple and natural set-theoretic models of pure lambda-calculus, providing a suitable framework for studying the lattice of lambda-theories.
In this talk, I will present the basic technical tool of partial graph models and Engeler's completions, and use it to show some closure properties of the class of “graph theories” (i.e. of lambda-theories arising from graph models): there exist a smallest graph theory, and a smallest sensible one (a sensible lambda-theory is one that equates all the unsolvables terms). Moreover, the biggest sensible graph theory is Böhm trees equality.
If time permits, I'll present also some work in progress, aiming to show that the minimal, sensible graph theory is H, the smallest sensible lambda-theory.