Pairs of Points in the Plane

Question. What is the largest $N\in\mathbb{N}$ so that given any $N$ pairs of points $$\{(a_j,b_j)\}_1^N\subset\mathbb{R}^2\times\mathbb{R}^2$$ there exist non-intersecting paths $f_j:I\to\mathbb{R}^2$ with $f_j(0)=a_j$ and $f_j(1)=b_j$?