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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 $T$?
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An universal theory is a theory axiomatised by a set of universal sentences, i.e. of the form $\forall x_1\forall x_2\ldots\forall x_n \varphi(x_1,\ldots,x_n)$ (where $\varphi$ is quantifier-free).

That, in addition to absence of function symbols or constants trivially implies that for every model of $T$, any nonempty subset (with inherited structure) is also a model of $T$.

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