Rich-bisector deletion bound ($\#\{\text{arcs with multiplicity at least }k\}=O(t{n}^2/k^2+tn\log n)$)

Every supporting centre of an arc between fixed endpoints lies on their perpendicular bisector. Multiplicity at least $k$ therefore makes that line contain at least $k$ of the original points. For each such line containing $s$ points, its $s$ centres each support at most $t$ circles; on one circle at most two consecutive-point arcs have that symmetry axis. The rich line bound summed over dyadic occupancies gives the displayed bound.