A chain is a totally ordered subset.

There is a question what a subset of P is. Let's say P is an order on a set A, i.e., . Then a subset of P could mean a subrelation on the same set A. However, if A is infinite, the premise of the statement is trivially false and so the statement is trivially true. Indeed, each finite subrelation has infinitely many isolated points, and each of them is a degenerate chain. So I think the problem is talking about finite subsets of A and restrictions of P on these subsets.

I have not worked out all details, but the idea may be as follows. Let C be a formula saying that for any k + 1 elements, some pair of them are comparable. We can prove that C holds for P.

Let A_n be a subset of A with n elements and let P_n be the restriction of P to A_n. By [P_n] I will denote a single formula that records the structure of P_n. It has constants c_1, ..., c_n and is a big conjunction of inequalities between these constants. If constants' names are chosen consistently, then [P_n] implies [P_k] for k < n. The problem statement implies that for each n, the set {[P_1], ..., [P_n], C} is consistent since P_n is its model. Therefore, the union of these sets is consistent by compactness theorem. A model of this union must include P as a substructure and so C is true on P.