Reconciling two statements of Ramsey's Theorem.

Question

In the appendix to Shelah's book on Classification Theory:

THEOREM 2.1 (Ramsey's Theorem): (1) For any infinite ordered set $I$, and $n$-place function $f$ from $I$, with range of cardinality $<\aleph_0$ there is an infinite set $J\subseteq I$ such that if $\bar s, \bar t\in^n J, \bar s, \bar t$ increasing, then $f(\bar s) = f(\bar t)$.

Wikipedia:

Let $X$ be some countably infinite set and colour the elements of $X^{(n)}$ (the subsets of $X$ of size $n$) in $c$ different colours. Then there exists some infinite subset $M$ of $X$ such that the size $n$ subsets of $M$ all have the same colour.

In the first statement it is assumed that the $n$-tuples are increasing, unlike in the second statement. Why is this? I have not worked through the proofs, but it would appear the first version is needlessly weakened. I find this hard to believe given the source!

Thanks

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Answer 1

Note that Wikipedia is talking about colouring $n$-element subsets of $X$ and Shelah is talking about colouring $n$-tuples in $X$. Shelah thus needs to pick a canonical way of viewing a finite subset as a tuple and an increasing enumeration is one such way.

To get from Shelah's version to Wiki's version, we view an $n$-element subset $A\subseteq X$ as an increasing $n$-tuple (in a unique way) and colour it accordingly. To get from Wiki's version to Shelah's, we colour each tuple according to its underlying set.

Answer 2

Ramsey's theorem is false for colorings of all the ordered $n$-tuples. Increasing is just a way to talk about restricting the coloring to subsets, where the theorem is true. Here it must mean strictly increasing, to avoid inconsistencies in the coloring of different presentations of a set by tuples with repeated elements.