There is Theorem by Eisenbud and Koh which can be stated as follows:

**Theorem.** Let $P$ be a set of points in $\mathbb{P}^r$, and let $d \ge 2$ be an integer. If, for all $k \ge 1$, no $dk + 2$ points of $P$ lie in a projective $k-$plane,
then $P$ impose independent conditions on forms of degree $d$; in fact, there is a multilinear form of degree $d$ containing any subset consisting of all but one of the
points, but missing the last.

Question.Do we need to consider $P$ to be a set of distinct points, or a locally complete intersection zero-dimensional subscheme will work, where the numbers in the theorem will be replaced by length?