Thin set theorems and cone avoidance

Abstract

The thin set theorem R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty ,\ell } asserts the existence, for every k k -coloring of the subsets of natural numbers of size n n , of an infinite set of natural numbers, all of whose subsets of size n n use at most ℓ \ell colors. Whenever ℓ = 1 \ell = 1 , the statement corresponds to Ramsey’s theorem. From a computational viewpoint, the thin set theorem admits a threshold phenomenon, in that whenever the number of colors ℓ \ell is sufficiently large with respect to the size n n of the tuples, the thin set theorem admits strong cone avoidance.

Let d 0 , d 1 , … d_0, d_1, \dots be the sequence of Catalan numbers. For n ≥ 1 n \geq 1 , R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty , \ell } admits strong cone avoidance if and only if ℓ ≥ d n \ell \geq d_n and cone avoidance if and only if ℓ ≥ d n − 1 \ell \geq d_{n-1} . We say that a set A A is R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty , \ell } -encodable if there is an instance of R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty , \ell } such that every solution computes A A . The R T > ∞ , ℓ n \operatorname {\mathsf {RT}}^n_{>\infty , \ell } -encodable sets are precisely the hyperarithmetic sets if and only if ℓ > 2 n − 1 \ell > 2^{n-1} , the arithmetic sets if and only if 2 n − 1 ≤ ℓ > d n 2^{n-1} \leq \ell > d_n , and the computable sets if and only if d n ≤ ℓ d_n \leq \ell .