Irregular pair of vertex sets

For disjoint nonempty vertex sets $U,V$ of a graph, an $\epsilon$-irregular pair is one that is not an $\epsilon$-regular pair of vertex sets. Equivalently there are witnesses $U^{\prime }\subseteq U$, $V^{\prime }\subseteq V$ with $|U^{\prime }|\ge \epsilon |U|$, $|V^{\prime }|\ge \epsilon |V|$ and $|d(U^{\prime },V^{\prime })-d(U,V)|>\epsilon$. Splitting both cells by these witnesses increases their contribution to the regularity energy by at least ${\epsilon }^4|U||V|/n^2$: the witness rectangle has relative weight at least ${\epsilon }^2$, its mean discrepancy has magnitude greater than $\epsilon$, and the Cauchy-Schwarz inequality bounds the weighted sum of squared discrepancies below by the rectangle weight times its squared mean. This is the local increment in the energy-increment proof of Szemerédi regularity.