9. APPLICATIONS OF COMPACTNESS
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(3) A subgraph (U, F ) of (V, E) is a clique if F = [U]2. (4) A subgraph (U, F ) of (V, E) is an independent set if F = ∅. That is, a graph is some collection of vertices, some of which are joined to one another. A subgraph is just a subset of the vertices, together with all edges joining vertices of this subset in the whole graph. It is a clique if it happens that the original graph joined every vertex in the subgraph to all other vertices in the subgraph, and an independent set if it happens that the original graph joined none of the vertices in the subgraph to each other. The question of when a graph must have a clique or independent set of a given size is of some interest in many applications, especially in dealing with colouring problems. Theorem 9.2 (Ramsey’s Theorem). For every n ≥ 1 there is an integer Rn such that any graph with at least Rn vertices has a clique with n vertices or an independent set with n vertices. Rn is the nth Ramsey number . It is easy to see that R1 = 1 and R2 = 2, but R3 is already 6, and Rn grows very quickly as a function of n thereafter. Ramsey’s Theorem is fairly hard to prove directly, but the corresponding result for infinite graphs is comparatively straightforward. Lemma 9.3. If (V, E) is a graph with infinitely many vertices, then it has an infinite clique or an infinite independent set. A relatively quick way to prove Ramsey’s Theorem is to first prove its infinite counterpart, Lemma 9.3, and then get Ramsey’s Theorem out of it by way of the Compactness Theorem. (If you’re an ambitious minimalist, you can try to do this using the Compactness Theorem for propositional logic instead!) Elementary equivalence and non-standard models. One of the common uses for the Compactness Theorem is to construct “nonstandard” models of the theories satisfied by various standard mathematical structures. Such a model satisfies all the same first-order sentences as the standard model, but differs from it in some way not expressible in the first-order language in question. This brings home one of the intrinsic limitations of first-order logic: it can’t always tell essentially different structures apart. Of course, we need to define just what constitutes essential difference. Definition 9.2. Suppose L is a first-order language and N and M are two structures for L. Then N and M are: (1) isomorphic, written as N ∼ = M, if there is a function F : |N| → |M| such that