**Motivation.** My sons participated in a large football tournament recently; everyone wanted to be in a team with everyone else at least once. Tricky!

**Formulation of the question.** For any positive integer $k\in\mathbb{N}$, let $\ \binom {\mathbb N} k\ $ denote the collection of $k$-element subsets of $\mathbb{N}.\ $ Let ${\cal I}$ be the collection of subsets of $\mathbb{N}$ that are both infinite and co-infinite (the complement is infinite).

Fix an integer $k\geq 2$. Is there a finite family $\ {\cal S}\subseteq {\cal I}\ $ such that for every $\ F\in\binom {\mathbb N} k\ $ there is $T\in{\cal S}\ $ with $F\subseteq T$ or $F\subseteq (\mathbb{N}\setminus T)$? If yes, what is the smallest size such a family ${\cal S}$ can have, in terms of $k$?

**Note.** I originally posted this question for $k=2$ and arbitrary subsets of $\mathbb{N}$, not just infinite/co-infinite ones. Zach Teitler quickly gave an elegant solution and suggested this generalization. First it seemed $k=2$ was solved, but it turns out $k=2$ is still unresolved.

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