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9. APPLICATIONS OF COMPACTNESS
We claim that every finite subset of Σ is satisfiable. The most direct way to verify this is to show how, given a finite subset ∆ of Σ, to produce a model M of ∆. Let n be the largest integer such that σn ∈ ∆ ∪ {σ2} (Why is there such an n?) and choose an integer k such that 2k ≥ n. Define a structure (G, ◦) for LG as follows: • G = { ha` | 1 ≤ ` ≤ ki | a` = 0 or 1 } • ha` | 1 ≤ ` ≤ ki ◦ hb` | 1 ≤ ` ≤ ki = ha` + b` (mod 2) | 1 ≤ ` ≤ ki That is, G is the set of binary sequences of length k and ◦ is coordinatewise addition modulo 2 of these sequences. It is easy to check that (G, ◦) is a commutative group with 2k elements in which every element has order 2. Hence (G, ◦) |= ∆, so ∆ is satisfiable. Since every finite subset of Σ is satisfiable, it follows by the Compactness Theorem that Σ is satisfiable. A model of Σ, however, must be an infinite commutative group in which every element is of order 2. (To be sure, it is quite easy to build such a group directly; for example, by using coordinatewise addition modulo 2 of infinite binary sequences.) Problem 9.1. Use the Compactness Theorem to show that there is an infinite (1) bipartite graph, (2) non-commutative group, and (3) field of characteristic 3, and also give concrete examples of such objects. Most applications of this method, including the ones above, are not really interesting: it is usually more valuable, and often easier, to directly construct examples of the infinite objects in question rather than just show such must exist. Sometimes, though, the technique can be used to obtain a non-trivial result more easily than by direct methods. We’ll use it to prove an important result from graph theory, Ramsey’s Theorem. Some definitions first: Definition 9.1. If X is a set, let the set of unordered pairs of elements of X be [X]2 = { {a, b} | a, b ∈ X and a 6= b }. (See Definition A.1.) (1) A graph is a pair (V, E) such that V is a non-empty set and E ⊆ [V ]2. Elements of V are called vertices of the graph and elements of E are called edges. (2) A subgraph of (V, E) is a pair (U, F ), where U ⊂ V and F = E ∩ [U]2 .