Results in mathematics whose only proof is model theoretic

Question

What are results in mathematics, for example in algebra, whose only proof so far used model theoretical arguments?

Answer 1

I don't know whether or not there is a non-model theory proof, but the first proof of the "unconditional" André-Oort conjecture for arbitrary products of modular curves was done by Pila using model theory (more explicitly, O-minimality).

Furthermore, there is a trend of results coming out right now linking model theory of graphs and combinatorics. For instance, see Regularity lemmas for stable graphs. Since much of this work is current and still on going, I would venture to say that this result (or results like this one) only have proofs in model theory (as of now).