I need to learn about Razborov's "flag algebras" (see http://bit.ly/1u1a1NB) to solve a problem about graphs. Flag algebras are a very general new algebraic tool for studying combinatorial structures. The theory of flag algebras uses some elementary model theory in its definition which is something I have not encountered before (in fact I have not taken any classes in formal logic beyond the first-year undergrad stuff). I've read through the first few sections of Marker's "Model Theory: An Introduction" (http://amzn.to/1zGuyKq) and I have easily understood the definitions and general approach; in particular, I think I only really need to understand the general concepts of languages $\mathcal{L}$, $\mathcal{L}$-structures, embeddings of $\mathcal{L}$-structures and some general facts about $\mathcal{L}$-theories and models, all of which seems fine.
Reading the beginning of Razborov's paper [p4], though, I found some terms that I have not seen defined which I'd like to clear up. In particular, he writes
"Let $T$ be a universal first-order theory with equality in a language $\mathcal{L}$ containing only predicate symbols; we assume that $T$ has infinite models. Our assumptions imply that every set of elements of a model of $T$ induces a model of $T$, and that $T$ has at least one finite model of every given size."
I have two essential questions:
- what does "universal" mean in this context?
- why do the assumptions imply that any subset of a model of $T$ induces a model of $T$?