Exercise 28, § 6

Let E be an infinite set and let $(\mathfrak{X}_i)_{1 \leqslant i \leqslant m}$ be a finite partition of the set $\mathfrak{F}_n(\mathrm{E})$ of subsets of E having $n$ elements. Show that there exists an index $i$ and an infinite subset F of E such that every subset of F with $n$ elements belongs to $\mathfrak{X}_i$. (Proof by induction on $n$. For each $a \in \mathrm{E}$ show that there exists an index $j(a)$ and an infinite subset $\mathrm{M}(a)$ of $\mathrm{E} - \{a\}$ such that, for every subset A of $\mathrm{M}(a)$ with $n-1$ elements, $\{a\} \cup \mathrm{A}$ belongs to $\mathfrak{X}_{j(a)}$. Then define a sequence $(a_i)$ of elements of E as follows : $a_1$ is an arbitrary element of E, $a_2$ is an arbitrary element of $\mathrm{M}(a_1)$, $a_3$ is defined in terms of $\mathrm{M}(a_1)$ and $a_2$ in the same way as $a_2$ was defined in terms of E and $a_1$, and so on. Show that the set F of elements of a suitable subsequence of the sequence $(a_i)$ satisfies the required conditions.)