First-order formula implying the existence of an infinite branch in a tree I want to find a formula which would imply the existence of an infinite branch in a tree. I know beforehand that my tree has countably many vertices (in fact, each vertex can possibly have a countable number of sons, so we cannot directly apply the Kőnig's lemma).
My question is: quantifying only over vertices (and edges, for that matter), is it possible to write a logical formula showing that the tree has an infinite branch (i.e. a tree satisfies the formula iff it has an infinite branch) ?
It is not hard to find one implying the existence of branches of arbitrary length, but this is not what I am looking for - although I am not even sure such a property can be expressed in first-order logic.
If this is impossible, are there some other assumptions that would need to be made in order to be able to write such a formula ? (For example, if the tree had a finite branching factor - which can be expressed by a first-order formula - then we could apply the Kőnig's lemma. As I already said, this is not the case here, but does there exist other hypothesis that would achieve the same result ?)

As it stands out, there is no easy way to answer my question. Indeed, one can find
here some sets of trees in term that are placed in the analytic hierarchy - rather than the arithmetic one, and I do not have any way of "discarding" those trees with the elements given in my question, meaning that the general problem is much harder that what can be dealt with FO-logic.
