Give a sentence having models of size $2n$ for every positive integer $n$ , but no finite models of odd size. (Here the language should include equality and will have whatever parameters you choose.) Suggestion: One method is to make a sentence that says, “Everything is either red or blue, and $f$ is a color-reversing permutation.”

Remark: Given a sentence $\sigma$ , it might have some finite models (i.e., models with finite universes). Define the spectrum of $\sigma$ to be the set of positive integers $n$ such that $\sigma$ has a model of size $n$ . This exercise shows that the set of even numbers is a spectrum.

For example if $\sigma$ is the conjunction of the field axioms (there are only finitely many, so we can take their conjunction), then its spectrum is the set of powers of primes. This fact is proved in any course on finite fields. The spectrum of $\neg\sigma$ , by contrast, is the set of all positive integers (non-fields come in all sizes). Günter Asser in 1955 raised the question: Is the complement of every spectrum a spectrum? Once you realize that simply taking a negation does not work (cf. the preceding paragraph), you see that this is a nontrivial question. In fact the problem, known as the spectrum problem, is still open. But modern work has tied it to another open problem, whether or not co-NP = NP.