Hrushovski construction
In model theory, a branch of mathematical logic, the Hrushovski construction generalizes the Fraïssé limit by working with a notion of "strong substructure" formula_1 rather than formula_2. It can be thought of as a kind of "model-theoretic forcing", where a (usually) stable structure is created, called the "generic" or "rich" model. The specifics of formula_1 determine various properties of the generic, with its geometric properties being of particular interest. It was initially used by Ehud Hrushovski to generate a stable structure with an "exotic" geometry, thereby refuting Zil'ber's Conjecture. Three conjectures. The initial applications of the  Hrushovski construction refuted two conjectures and answered a third question in the negative. Specifically, we have: The construction. Let "L" be a finite relational language. Fix C a class of "finite" "L"-structures which are closed under isomorphisms and substructures. We want to strengthen the notion of substructure; let formula_1 be a relation on pairs from C satisfying: Definition. An embedding formula_22 is "strong" if formula_23 Definition. The pair formula_24 has the "amalgamation property" if formula_25 then there is a formula_26 so that each formula_27 embeds strongly into formula_28 with the same image for formula_29 Definition. For infinite formula_28 and formula_31  we say formula_32 iff formula_33 for formula_34 Definition. For any formula_35 the "closure" of formula_36 in formula_37 denoted by formula_38 is the smallest superset of formula_36 satisfying formula_40 Definition. A countable structure formula_41 is "formula_24-generic" if: Theorem. If formula_51 has the amalgamation property, then there is a unique formula_51-generic. The existence proof proceeds in imitation of the existence proof for Fraïssé limits. The uniqueness proof comes from an easy back and forth argument.