  The thin set theorem $\mathsf{RT}^n_{<\infty,\ell}$ asserts the existence,
for every $k$-coloring of the subsets of natural numbers of size $n$, of an
infinite set of natural numbers, all of whose subsets of size $n$ use at most
$\ell$ colors. Whenever $\ell = 1$, 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 $\ell$ is
sufficiently large with respect to the size $n$ of the tuples, then the thin
set theorem admits strong cone avoidance.
  Let $d_0, d_1, \dots$ be the sequence of Catalan numbers. For $n \geq 1$,
$\mathsf{RT}^n_{<\infty, \ell}$ admits strong cone avoidance if and only if
$\ell \geq d_n$ and cone avoidance if and only if $\ell \geq d_{n-1}$. We say
that a set $A$ is $\mathsf{RT}^n_{<\infty, \ell}$-encodable if there is an
instance of $\mathsf{RT}^n_{<\infty, \ell}$ such that every solution computes
$A$. The $\mathsf{RT}^n_{<\infty, \ell}$-encodable sets are precisely the
hyperarithmetic sets if and only if $\ell < 2^{n-1}$, the arithmetic sets if
and only if $2^{n-1} \leq \ell < d_n$, and the computable sets if and only if
$d_n \leq \ell$.
