Given a positive integer $u$, how many $k$-dimensional vectors whose coordinates are all in $\lbrace 1, 2, 3, ..., u\rbrace$ can you choose so that any $k$ of them are linearly independent? Equivalently, what is the size of the largest subset of $\lbrace 1, 2, 3, ... u \rbrace^k$ so that each hyperplane through the origin contains at most $k-1$ of them?

If $k=2$, two vectors are linearly dependent iff they have the same slope, so the maximum number of pairwise independent vectors is the number of distinct slopes $y/x$ with $1\le x,y \le u$,

$$ -1 + 2\sum_{n=1}^u \phi(n),$$

since the number of slopes up to $1$ with reduced denominator $n$ is $\phi(n)$, and slopes other than $1$ come in reciprocal pairs.

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