Given a well-ordering of $\mathbb{R}$, there is a natural way to define an associated partition of pairs of real numbers into two pieces: one assigns the value $0$ to a pair $r<s$ if the well-ordering agrees with the standard ordering on the pair, and gives it the value $1$ if not.

This construction is due originally to Sierpiński, and it is an important example/counterexample in Ramsey Theory because this coloring cannot have an uncountable homogeneous set: the separability of $\mathbb{R}$ tells us there must be a rational number between any two consecutive elements of a well-ordered (or reverse well-ordered) subset of $\mathbb{R}$.

In the square-bracket notation (discussed in my last question), Sierpiński's result says $$2^{\aleph_0}\nrightarrow [\aleph_1]^2_2.$$

Now Galvin and Shelah state the following in their paper [1]

We remark that an easy generalization of Sierpiński's proof shows that $2^{\aleph_0}\nrightarrow [\aleph_1]^r_{r!(r-1)!}$ for every positive integer r.

Their conclusion says that one can color the increasing $r$-tuples of reals with $r!(r-1)!$ colors in such a way that for any uncountable $X\subseteq\mathbb{R}$ each of these colors is realized by some $r$-tuple drawn from $X$.

I was unsuccessful at verifying their claim, as the most straightforward (to my mind) generalization of Sierpiński's proof provides a weaker coloring of $r$-tuples using $r!$-colors: one takes a well-ordering $<^*$ of $\mathbb{R}$ and the coloring assigns to an $r$-tuple $a_0<\dots<a_{r-1}$ (in the usual ordering) the permutation of the index set $r$ that arises in the natural way once you rewrite your $r$-tuple in $<^*$-increasing order. Once again, the separability of finite powers of $\mathbb{R}$ is what allows you to conclude that every color appears on every uncountable set. So, this argument establishes only $$2^{\aleph_0}\nrightarrow [\aleph_1]^r_{r!}$$ and my question is how does one improve to obtain the stronger result mentioned by Galvin and Shelah?

It is not out of the question that there is a typo in their paper, but I am also aware that their definition of ''easy generalization'' is probably not the same as my own.

[1] *Galvin, Fred; Shelah, Saharon*, **Some counterexamples in the partition calculus**, J. Comb. Theory, Ser. A 15, 167-174 (1973). ZBL0267.04006.