Fixpoint zip {X} (f1 f2 : nat -> X) n :=
  match n with
  | O => f1 O
  | S O => f2 0
  | S (S n) => zip (fun n => f1 (S n)) (fun n => f2 (S n)) n
  end.

Lemma ramsey_inf {X} (A : X -> Prop) (P : nat -> X -> Prop) : (forall P, P \/ ~P) -> finite A -> infinite (fun n => exists x, A x /\ P n x) -> exists x, A x /\ infinite (fun n => P n x).