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

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 $I(P,\mathcal{C})=O_k(n^{(3k+1)\mathrm{/}(2k+1)})$. 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 $I(P,\mathcal{C})=o(n^{(3k+1)\mathrm{/}(2k+1)})$. In the special case of pseudo-segments, this extends Solymosi’s theorem on dense point-line arrangements to dense arrangements of pseudo-segments.