Many empty triangles have a common edge

Given a finite point set $X$ in the plane, the degree of a pair ${x,y} \subset X$isthenumberofemptytriangles$t=conv{x,y,z}$, where empty means $t\cap X=\{x,y,z\}$. Define $degX$ as the maximal degree of a pair in $X$. Our main result is that if $X$ is a random sample of $n$ independent and uniform points from a fixed convex body, then $degX\ge cn\mathrm{/}\ln n$ in expectation.