Does every two-set partition of a projective plane contain a line?

Question

Define a two-set partition of a projective plane as a partition of the points into two sets. Does there exist for any two-set partition a set in the partition that contains a line? What about infinite projective planes?

Answer 1

Every projective plane of order at least $3$ (i.e. at least $4$ points on a line) admits a "$2$-set partition" in which neither set contains a line. In other words, we can color each point red or blue so that every line contains at least one point of each color. (As you probably know, the projective plane of order $2$ does not admit such a partition.)

Proof. Choose three non-concurrent lines $\ell_1$, $\ell_2$, and $\ell_3$. Color a point red if it lies on exactly one of those three lines, blue otherwese.

I claim that every line contains points of both colors. This is plainly true for the lines $\ell_1$, $\ell_2$, and $\ell_3$, so consider a line $\ell\notin\{\ell_1,\ell_2,\ell_3\}$. Then $\ell$ must contain a red point, since it can't intersect all three of the lines $\ell_1$, $\ell_2$, $\ell_3$ in blue points. On the other hand, since every red point lies on $\ell_1$ or $\ell_2$ or $\ell_3$, and since $\ell$ intersects each of those three lines in just one point, there are at most $3$ red points on $\ell$, so there is at least one blue point on $\ell$.