 Jockusch triple coloring computing the halting set

Let $K_s$ be a computable increasing finite-stage enumeration of the halting problem set $K$. Color $x<y<z$ by whether the approximations below $x$ differ at stages $y,z$. This is a recursive two-color partition of triples. An infinite homogeneous set cannot have color one: for fixed $x$, the finite approximation below $x$ eventually stabilizes. If its color is zero, choose $x$ from that set above an input $n$, then another member $y>x$; agreement with arbitrarily large later stages gives $K(n)=K_y(n)$. Thus every infinite homogeneous set computes $K$, and none is a recursive set. This makes the failure of an effective version of Ramsey's theorem explicit.