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For the question bellow I will use notations used in Enderton's book.

Let $\mathcal{L}$ be a language with $=$, $\forall$ and $R$ a binary relation symbol. A model $\mathfrak{A}$ is called a graph if $\mathfrak{A}$ satisfies

  • $\forall x (\neg x R x)) $ (interpreted as no vertex is connected to itself via an edge)
  • $\forall x \forall y (xRy \to yRx)$ (interpreted as undirected graphs)

A clique in a graph $\mathfrak{A}$ is a set $\{v_1, v_2, \ldots v_n\} \subseteq |\mathfrak{A}|$ of $n$ distinct elements such that for all $i\neq j$ we have that $(v_1,v_j) \in R^{\mathfrak{A}}$. (interpreted as a clique of size $n$ is, $n$ distinct vertices such that each vertex is connected to others via an edge).

With these definitions I am given to prove that, there is no set of wff $\Sigma$ (finite or infinite) in the language $\mathcal{L}$ such that $\mathfrak{A}\vDash\Sigma$ if and only if $\mathfrak{A}$ is finite with every clique is contained in a clique of even size.

Guest_000
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It is a fact that if an $\mathrm{EC}_\Delta$ class (more commonly called an elementary class, or a first-order axiomatizable class, or the class of models of a first-order theory) contains arbitrarily large finite structures, then it contain an infinite structure. Two proofs are given in the answers to this question, and you can find many more explanations by searching for "compactness arbitrarily large finite models" on this site.

Now it follows immediately that the class of finite graphs such that every maximal clique has even size is not $\mathrm{EC}_\Delta$, since this class contains arbitrarily large graphs (e.g. the finite complete graphs of even size) but no infinite graphs.

Alex Kruckman
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