Motivation. My eldest son starts school tomorrow. His class is split in two groups of $10$ students each. From time to time, the groups are rearranged. I wondered how many rearrangements are needed until every student has been in the same group with every other student at least once. This led to a more general question.

Question. Is there a positive integer $n$ and non-constant functions $f_i: \omega \to \{0,1\}$ for $i \in \{0,\ldots,n-1\}$ with the following property?

For all $a,b\in \omega$ there is $i \in \{0,\ldots,n-1\}$ such that $f_i(a) = f_i(b)$.

If yes, how small can $n$ be?

Motivation. My eldest son starts school tomorrow. His class is split in two groups of $10$ students each. From time to time, the groups are rearranged. I wondered how many rearrangements are needed until every student has been in the same group with every other student at least once. This led to a more general question.

Question. $f: \omega \to \{0,1\}$ is said to be fair if the preimages $f^{-1}(\{0\})$ and $f^{-1}(\{1\})$ are both infinite. Is there a positive integer $n$ and fair functions $f_i: \omega \to \{0,1\}$ for $i \in \{0,\ldots,n-1\}$ with the following property?

For all $a,b\in \omega$ there is $i \in \{0,\ldots,n-1\}$ such that $f_i(a) = f_i(b)$.

If yes, how small can $n$ be?