Finite graphs where every maximal clique is of even size is not EC$_\Delta$

Question

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 the class of finite graphs where every maximal clique of even size is not EC$_\Delta$ in the sense defined in Enderton's logic book. That is to say 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.

This question is the last part of the following question with parts:

1. For each $n\geq 2$ find a wff $\phi_n(x_1, \ldots x_n)$ with $n$ free variables such that $\mathfrak{A}\vDash \phi_n[[v_1\ldots v_n]]$ iff $\{v_1\ldots v_n\}$ is a clique of size $n$.
2. For each $n$ find a sentence $\psi_n$ such that $\mathfrak{A}\vDash \psi_n$ iff ever clique of size $n$ is contained in a clique of size $n+1$.
3. Find a set of sentences $\Lambda$ such that $\mathfrak{A}\vDash \Lambda$ iff every clique of odd size is contained in a clique of even size.
4. Prove that class of finite graphs with every clique is contained in a clique of even size is not EC$_\Delta$

There is a hint given.

Hint: Add infinitely many distinct, new constant symbols $c_1, c_2,\ldots $ to $\mathcal{L}$ to get a new language $\mathcal{L}'$. Use the compactness theorem on an appropriate set $\Gamma$ of sentences in $\mathcal{L}'$ with $\Gamma\supseteq\Sigma$.

PS: This is a past exam question from a course that I am currently taking. I was able to do the first 3 parts. I understand that for the last part the general idea is to show that every finite subset of $\Gamma$ above is satisfiable and therefore by compactness theorem $\Lambda$ is satisfiable. But a graph that satisfies $\Gamma$ is finite but then to show that such a graph must also be infinite giving us a contradiction. I am unsure on how to specifically use the hint to obtain this.

Answer 1

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.