description

Name: Ramsey
Title: Ramsey Theory
Author: Marc Bezem
Institution: Utrecht University 
Description:
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.  

Keywords: dimension one Ramsey theorem, constructive mathematics,
almost full sets
Category: Mathematics/Logic/See also
Category: Mathematics/Combinatorics and Graph Theory
Category: Miscellaneous/Extracted Programs/Combinatorics