Extremal structure in dense arrangements of $k$-intersecting curves

TL;DR

This paper improves bounds on incidences between points and $k$-intersecting curves in the plane by excluding certain local patterns, extending previous results to pseudo-segments.

Contribution

It establishes a new upper bound for incidences when specific local incidence patterns are absent, generalizing Solymosi's theorem to pseudo-segments.

Findings

Improved incidence bounds under pattern exclusion

Extension of Solymosi's theorem to pseudo-segments

Demonstrates that certain local patterns influence the maximum incidence count

Abstract

Let $P$ be a set of $n$ points in the plane, and let $\mathcal C$ be a collection of $n$ simple $k$-intersecting curves, meaning that every two distinct curves of $\mathcal C$ meet in at most $k$ points. A classical theorem of Pach and Sharir from 1998 gives the upper bound $I(P,\mathcal C)=O_k(n^{(3k+1)/(2k+1)})$. We prove that this bound can be improved when one excludes a complete local incidence pattern. More precisely, for any fixed integers $s>k+1\ge 2$, if there do not exist $s$ points of $P$ such that every $(k+1)$-tuple among them is contained in a distinct curve of $\mathcal C$, then $I(P,\mathcal C)=o(n^{(3k+1)/(2k+1)})$. In the special case of pseudo-segments, this extends Solymosi's theorem on dense point-line arrangements to dense arrangements of pseudo-segments.