Let KL denote König's Lemma (for trees over $\mathbb{N}$), and RT(3) denote the
Infinite Ramsey Theorem for triples over $\mathbb{N}$ (notation as in Simpson's
book *Subsystems of second order arithmetic*, 2ed, 2009, see definitions below).

Question:Is there a simple direct derivation of KL from RT(3)?

KL, RT(3), and ACA$_0$ are equivalent over RCA$_0$, and Simpson's book has proofs (in RCA$_0$) of the following implications: ACA$_0$ $\implies$ KL $\implies$ RT(3) $\implies$ ACA$_0$.

I am
looking for a direct "simple" proof of RT(3) $\implies$ KL (formalizable
in RCA$_0$ and *without* having to go through an intermediate ACA$_0$).

## Definitions

Here KL (König's Lemma) is the statement "Every finitely branching infinite tree over $\mathbb{N}$ has an infinite branch", where (a) $T$

*is a tree over*$\mathbb{N}$ if $T$ is a set of finite sequences from $\mathbb{N}$ such that any initial segment of any sequence in $T$ is in $T$, (b) $T$*is finitely branching*if for any sequence $u \in T$ there are at most finitely many sequences in $T$ of length $\operatorname{length}(u)+1$ which extend $u$, and (c) a*branch of a tree*is a maximal linearly ordered subset of it (under the relation $u \preceq v$ iff $u$ is an initial segment of $v$).RT(3) (Infinite Ramsey Theorem for triples) is the statement "If $F$ is a finite set and $f \colon [\mathbf{N}]^3 \to F$, then there is an infinite subset $H$ of $\mathbb{N}$ such that $f$ is constant on $[H]^3$", where $[X]^n$ denotes the set of all $n$-element subsets of $X$ (for any set $X$).