Let $\def\A{\mathbb A}\def\F{\mathbb F}\F_3$ be the Galois field with three elements and let $\A^d=\A^d(\F_3)$ be the affine space of dimension $d$ over $\mathbb F_3$ —the subject is combinatorics and not algebraic geometry so this is essentially just the set $\F_3^d$ with the incidence structure whose blocks are the affine subspaces.

Suppose $p\in\A^d$ is point and $X\subseteq\A^d$ is a set not containing $p$ which intersects every line through $p$ in exactly one point.

Is it true that $X$ contains an affine subspace of dimension $d-1$?

If this is true, then in fact $X$ is a disjoint union $P_0\cup P_1\cup\cdots\cup P_{d-1}$ where each $P_i$ is an affine subspace of $\A^d$ of dimension $i$, and this decomposition is unique.

I *think* that I have a (horrid) proof of this but I know that there are people who are immensely more fluent in this sort of thing than I...

isof the form $P_0\cup\dots\cup P_{d−1}$, and just replace one of the points of the $P_{d−1}$ with the other point on the same line through $p$. – Jeremy Rickard May 10 '13 at 11:15