Let $K$ be a finite field (say $K = \mathbb{F}_q$), and $S$ a subset of $K P^2$ such that $$\forall g \in \mathbf{PGL}(3,K), g.S \cap S \neq \emptyset.$$ Can such an $S$ be smaller (i.e. have strictly less elements) than a projective line, namely $\vert S \vert \leqslant q$ ?
EDIT : I originally wrote the question in $KP^n$, this can still be raised, but a line should certainly be replaced by a larger set in that case (e.g. a subspace of half dimension if $n$ is even). Nevertheless, the case $n=2$ is the starting point.
NB: This question can be made much more general (in fact as soon as one encounters a group action that preserves some measure) ; any help on finding appropriate terms or keywords for this kinds of problem in the litterature will be useful for me.