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Given a set of $n^{r}$ points $X_{r} = \{ x_{1}, \cdots, x_{n^{r}} \}$ occupying a codim $t^{r}$ subspace in $\mathbb{R}^{n^{r}}$. Let $M_{r}$ be the set of $t^{r}$-tuples of these points.. So $|M_{r}| = \frac{n^{r}!}{(n^{r}-t^{r})!t^{r}!}$. Each $m \in M_{r}$ lie in a dim $t^{r}-1$ hyperplane in $\mathbb{R}^{n^{r}-t^{r}}$. So $M_{r}$ can be thought of as a set of dim $t^{r}-1$ hyperplanes. Consider $M_{sub,r}$ a subset of $M_{r}$. What is the maximum number of hyperplanes in $M_{sub,r}$ such that their intersection is empty?

In my case there is a particular tensor structure to $M_{sub,r}$. That is $M_{sub,r}$ is just the $r$-fold tensor product of $M_{sub,1}$ and $|M_{sub,1}| = n$. Also $X_{r}$ is $n$-fold tensor product of $X$.

Are there any known results or techniques to study this type of problem?

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