What is the maximum number of $k$-tuples($3\le k\le n$) of $n$ numbers such that no pair is repeated in any of the tuples?
The maximum of number of $k$-tuples occur when $k=\lfloor\frac{n}{2}\rfloor$ as $n\choose \lfloor\frac{n}{2}\rfloor$ is maximum among all binomial coefficients. But, when we give the additional constraint that no pair is repeated twice, I have a hard time in getting the answer. For example, for $n=6$, I think the answer is $4$: the tuples desired are $(123),(145),(246),(356)$. Is it possible that the maximum in this case occurs for $k\neq\lfloor\frac{n}{2}\rfloor$? Any hints? Thanks beforehand.
