Small-cell deletion in weighted regularity ($|V_i|<\epsilon n/M⟹\text{at most }\epsilon n\text{ deleted vertices}$)

A partition with at most $M$ cells has at most $\epsilon n$ vertices in cells smaller than $\epsilon n/M$. Deleting all edges incident to those vertices costs at most $\epsilon n^2$ edges. Every remaining cell has a fixed positive fraction of all vertices, which permits a regular triangle counting lemma to give a cubic count without requiring equal cell sizes.