The initial applications of the Hrushovski construction refuted two conjectures and answered a third question in the negative. Specifically, we have:

• Lachlan's Conjecture. Any stable ${\displaystyle \aleph _{0}}$ -categorical theory is totally transcendental.^[2]

• Zil'ber's Conjecture. Any uncountably categorical theory is either locally modular or interprets an algebraically closed field.^[3]

• Cherlin's Question. Is there a maximal (with respect to expansions) strongly minimal set?

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 ${\displaystyle \leq }$  be a relation on pairs from C satisfying:

• ${\displaystyle A\leq B}$  implies ${\displaystyle A\subseteq B.}$ 
• ${\displaystyle A\subseteq B\subseteq C}$  and ${\displaystyle A\leq C}$  implies ${\displaystyle A\leq B}$ 
• ${\displaystyle \varnothing \leq A}$  for all ${\displaystyle A\in \mathbf {C} .}$ 
• ${\displaystyle A\leq B}$  implies ${\displaystyle A\cap C\leq B\cap C}$  for all ${\displaystyle C\in \mathbf {C} .}$ 
• If ${\displaystyle f\colon A\to A'}$  is an isomorphism and ${\displaystyle A\leq B}$ , then ${\displaystyle f}$  extends to an isomorphism ${\displaystyle B\to B'}$  for some superset of ${\displaystyle B}$  with ${\displaystyle A'\leq B'.}$ 

Definition. An embedding ${\displaystyle f:A\hookrightarrow D}$  is strong if ${\displaystyle f(A)\leq D.}$ 

Definition. The pair ${\displaystyle (\mathbf {C} ,\leq )}$  has the amalgamation property if ${\displaystyle A\leq B_{1},B_{2}}$  then there is a ${\displaystyle D\in \mathbf {C} }$  so that each ${\displaystyle B_{i}}$  embeds strongly into ${\displaystyle D}$  with the same image for ${\displaystyle A.}$ 

Definition. For infinite ${\displaystyle D}$  and ${\displaystyle A\in \mathbf {C} ,}$  we say ${\displaystyle A\leq D}$  iff ${\displaystyle A\leq X}$  for ${\displaystyle A\subseteq X\subseteq D,X\in \mathbf {C} .}$ 

Definition. For any ${\displaystyle A\subseteq D,}$  the closure of ${\displaystyle A}$  in ${\displaystyle D,}$  denoted by ${\displaystyle \operatorname {cl} _{D}(A),}$  is the smallest superset of ${\displaystyle A}$  satisfying ${\displaystyle \operatorname {cl} (A)\leq D.}$ 

Definition. A countable structure ${\displaystyle G}$  is ${\displaystyle (\mathbf {C} ,\leq )}$ -generic if:

• For ${\displaystyle A\subseteq _{\omega }G,A\in \mathbf {C} .}$ 

• For ${\displaystyle A\leq G,}$  if ${\displaystyle A\leq B}$  then there is a strong embedding of ${\displaystyle B}$  into ${\displaystyle G}$  over ${\displaystyle A.}$ 

• ${\displaystyle G}$  has finite closures: for every ${\displaystyle A\subseteq _{\omega }G,\operatorname {cl} _{G}(A)}$  is finite.

Theorem. If ${\displaystyle (\mathbf {C} ,\leq )}$  has the amalgamation property, then there is a unique ${\displaystyle (\mathbf {C} ,\leq )}$ -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.