  The purpose is to study the strength of Ramsey's Theorem for pairs restricted
to recursive assignments of $k$-many colors, with respect to Intuitionistic
Heyting Arithmetic. We prove that for every natural number $k \geq 2$, Ramsey's
Theorem for pairs and recursive assignments of $k$ colors is equivalent to the
Limited Lesser Principle of Omniscience for $\Sigma^0_3$ formulas over Heyting
Arithmetic. Alternatively, the same theorem over intuitionistic arithmetic is
equivalent to: for every recursively enumerable infinite $k$-ary tree there is
some $i < k$ and some branch with infinitely many children of index $i$.
