Compactness
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This document provides definitions and concepts related to logic, languages, structures, and the compactness theorem. It defines logical symbols and objects for propositional and first-order logic. A first-order language consists of function symbols, relation symbols, and constant symbols. A structure assigns interpretations to these symbols. The compactness theorem states that a theory is satisfiable if and only if every finite subset of the theory is satisfiable.
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It should be noted that an edge-joint partition of a graph is some partition of a graph's edges such that if an edge, e (between vertex a and b), is in a partition then if for any vertex c not equal to a or b, c is in the neighborhood of a AND b then ac and bc are also in the same partition as e. If a graph is simple this is the same as saying that if an edge is in a partition any K_3's that contain this edge are also in said partition.
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