Is there a sentence in the language $\{xRy\}$ with only infinite models?

Question

Can you find a sentence in a language with only a binary relation $R$, all of whose models are infinite?

Answer 1

Hint: What if $R$ represents a linear order with no maximum element?

Answer 2

Hint: you can express “$R$ is a dense linear order” in a single sentence. Or even “R is a dense partial order”.

Answer 3

Arthur and tomasz gave very good and generally canonical examples. Here is another one.

Write the conjunction of the following sentences:

1. $R$ is a function.
2. The domain of $R$ is the entire universe.
3. The range of $R$ is not the entire universe.
4. $R$ is injective.