Balanced intersection size distributions in projective planes

Abstract

Given a point set $S$ in a projective plane ${\hat{I}}_q$ of order $q$, each line $\ell$ determines a secant size $|S\cap \ell |$. We study how balanced the secant-size distribution can be for the line set $\mathcal{L}$ of the plane, in other words, how many lines must share the same secant size. We show that ${\min }_{S\subseteq {\hat{I}}_q}{\max }_k|\{\ell \in \mathcal{L}:|\ell \cap S|=k\}|=\hat{I}˜(q^{3/2}).$ This shows a large contrast with the case of real projective (or affine) plane, where ${\max }_{k>1}|\{\ell \in \mathcal{L}:|\ell \cap S|=k\}|$ is always at least the third of $|\{\ell \in \mathcal{L}:|\ell \cap S|>1\}|$. We also discuss explicit constructions in addition to randomized point sets, that are asymptotically close to be optimal, and point out a link between the constructions and character-sum estimates. Finally, we explore the relation between balanced secant size distributions and legitimate colorings, studied by Alon and Füredi, and prove a result that might resemble the Erdős-Faber-Lovász conjecture.