I am not sure how to even start solving this problem. I tried to solve it using the semantic tree, but with no success. It’s an exercise from the book Mathematical Logic for Computer Science by Mordechai Ben-Ari, page 154, exercise 7.10.
Prove that the following formula has no finite models: ∀x∃yp(x,y)∧∀x¬p(x,x)∧∀x∀y∀z(p(x,y)∧p(y,z)→p(x,z)).
Note that $p$ is transitive and irreflexive. Hence, it is also asymmetric, and thus a strict partial order. This means that all objects can be ordered accordingly. Moreover, given any finite number of objects, that means that there is a 'smallest' element, i.e. an element $x$ such that for no $y$: $p(x,y)$. However, that contradicts the $\forall x \exists y p(x,y)$. So, the domain cannot be finite.