We prove that for every $ε>0$ there exists $δ>0$ such that the following holds. Let $\mathcal{C}$ be a collection of $n$ curves in the plane such that there are at most $(\frac{1}{4}-ε)\frac{n^{2}}{2}$ pairs of curves $\{α,β\}$ in $\mathcal{C}$ having a nonempty intersection. Then $\mathcal{C}$ contains two disjoint subsets $\mathcal{A}$ and $\mathcal{B}$ such that $|\mathcal{A}|=|\mathcal{B}|\geq δn$, and every $α\in \mathcal{A}$ is disjoint from every $β\in\mathcal{B}$. On the other hand, for every positive integer $n$ there exists a collection $\mathcal{C}$ of $n$ curves in the plane such that there at most $(\frac{1}{4}+ε)\frac{n^{2}}{2}$ pairs of curves $\{α,β\}$ having a nonempty intersection, but if $\mathcal{A},\mathcal{B}\subset \mathcal{C}$ are such that $|\mathcal{A}|=|\mathcal{B}|$ and $α\cap β=\emptyset$ for every $(α,β)\in \mathcal{A}\times\mathcal{B}$, then $|\mathcal{A}|=|\mathcal{B}|=O(\frac{1}ε\log n)$.