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$\phi_n(x,y,z)\in FO$, we can use only three variables, but we can requantify them. Only available relation is edge relation $E$. For each finite directed cycle: $$(C, x : a, y : b, z : c) \models \phi_n(x,y,z)\leftrightarrow \text{$C$ has $3n$ edges, directed distances: dist(a,b) = dist(b,c) = dist(c,a) = n}$$

So, lets look at my trial:
$\phi_n(x,y,z) \equiv \psi_{n}(x,y)\wedge \psi_{n}(y,z) \wedge \psi_{n}(z,x)$

$\psi_0(x,y) \equiv x = y$
$\psi_n(x,y) \equiv \exists!_z(E(x,z) \wedge \psi_{n-1}(z,y))$

$\exists!$ means that there exists exactly one. Can someone check my solution ?

1 Answers 1

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Your question is not clear; what is $n$? Assuming that $n$ is a natural number (in the meta-system), your idea is roughly correct, but there are two issues:

  1. You did not say whether your first-order logic allows "$\exists!$" as an inbuilt notation. Otherwise you would have to use only ordinary quantifiers. It is possible, and I will leave you to figure out how to do it. Hint below.

  2. You should be more precise what your formulae are, so as to make clear what you do with the variables. Otherwise you would not have proven that it can be done using only $3$ variables.

Hint:

We did this kind of axiomatization before. Although it cannot force the graph to be finite, it can force enough that you can handle the $3n$-node constraint separately using your idea.

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    For those who don't know what we "did before", here's the complete hint (but don't look if you want to try yourself!): Find an axiomatization of a larger class of graphs that contains the desired class (of $3n$-cycles) but can be described 'locally' (namely using fixed number of quantifiers independent of $n$). – 2017-02-12