Ramsey Theory. For dimension one, the Infinite Ramsey Theorem states that, for any subset A of the natural numbers nat, either A or nat\A is infinite. This special case of the Pigeon Hole Principle is classically equivalent to: if A and B are both co-finite, then so is their intersection. None of these principles is constructively valid. In [VB] the notion of an almost full set is introduced, classically equivalent to co-finiteness, for which closure under finite intersection can be proved constructively. A is almost full if for every (strictly) increasing sequence f: nat -> nat there exists an x in nat such that f(x) in A. The notion of almost full and its closure under finite intersection are generalized to all finite dimensions, yielding constructive Ramsey Theorems. The proofs for dimension two and higher essentially use Brouwer's Bar Theorem. In the proof development below we strengthen the notion of almost full for dimension one in the following sense. A: nat -> Prop is called Y-full if for every (strictly) increasing sequence f: nat -> nat we have (A (f (Y f))). Here of course Y : (nat -> nat) -> nat. Given YA-full A and YB-full B we construct X from YA and YB such that the intersection of A and B is X-full. This is essentially [VB, Th. 5.4], but now it can be done without using axioms, using only inductive types. The generalization to higher dimensions will be much more difficult and is not pursued here.
opam install coq-ramsey.8.6.0