1. Show that it can always be done for $N3$ points on a plane of dimension $d2$, unless they are collinear.

2. Show that it cannot always be done for $N4$ points on a plane of dimension $d2$.

3. Show that it can always be done for $N4$ points in a space of dimension $d3$, unless they are coplanar.

4. Show that it cannot always be done for $N5$ points in a space of dimension $d3$.

5. The ambitious student may wish to prove that $N$ points in general position (but not $N+1$) are linearly separable in a space of dimension $N-1$.

Consider the problem of separating
$N$ data points into positive and negative examples using a linear
separator. Clearly, this can always be done for $N2$ points
on a line of dimension $d1$, regardless of how the points are
labeled or where they are located (unless the points are in the same
place).

1. Show that it can always be done for $N3$ points on a
plane of dimension $d2$, unless they are collinear.

2. Show that it cannot always be done for $N4$ points on a
plane of dimension $d2$.

3. Show that it can always be done for $N4$ points in a
space of dimension $d3$, unless they are coplanar.

4. Show that it cannot always be done for $N5$ points in a
space of dimension $d3$.

5. The ambitious student may wish to prove that $N$ points in general
position (but not $N+1$) are linearly separable in a space of
dimension $N-1$.