Bootstrapping the primitive recursive functions by only 27 colors

A coloring of the 3-element subsets of N with 27 colors is given such that if {s0, s1, s2, …, sr} (r ⩾ s0 − 1) is homogeneous, then Fω(si) < si+1 (i < τ) where Fω is an Ackermann-like function.