Exercise 27, § 6

Let L be an infinite set and let $(\mathrm{E}_\lambda)_{\lambda \in \mathbf{L}}$ be a family of sets indexed by L. Suppose that for each integer $n > 0$ the set of $\lambda \in \mathrm{L}$ such that Card $(\mathrm{E}_\lambda) > n$ is equipotent to L. Show that there exists a subset F of the product $\mathrm{E} = \prod_{\lambda \in \mathbf{L}} \mathrm{E}_\lambda$, such that Card $(\mathrm{F}) = 2^{\mathrm{Card\ (L)}}$, and such that F has the following property : for each finite sequence $(f_k)_{1 \leqslant k \leqslant n}$ of distinct elements of F there exists $\lambda \in \mathrm{L}$ such that the elements

are all distinct. (Show first that there exists a partition $(\mathrm{L}_j)_{j \in \mathbf{N}}$ of L such that Card $(\mathrm{L}_j) =$ Card (L) for all $j$, and such that Card $(\mathrm{E}_\lambda) \geqslant 2^j$ for each $\lambda \in \mathrm{L}_j$. Hence reduce to the case where L is the sum of the countable family of sets $\mathrm{X}^j$ $(j \geqslant 1)$, where X is an infinite set, and $\mathrm{E}_\lambda = 2^j$ for each $\lambda \in \mathrm{X}^j$. With each mapping $g \in 2^{\mathrm{X}}$ of X into 2, associate the element $f \in \mathrm{E}$ such that $f(\lambda) = (g(x_1), \ \ldots, \ g(x_j))$ whenever $\lambda = (x_k)_{1 \leqslant k \leqslant j} \in \mathrm{X}^j$; show that the set F of elements $f \in \mathrm{E}$ so defined has the required property.)