Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 147 / 3 / i / Solution

For nonempty $A\subseteq X$ and $B\subseteq Y$, their edge density of a bipartite graph isThe pair $(A,B)$ is $\epsilon$-regular when every $A^{\prime }\subseteq A$ and $B^{\prime }\subseteq B$ with $|A^{\prime }|\ge \epsilon |A|$ and $|B^{\prime }|\ge \epsilon |B|$ satisfyFor set partitions $X=X_1\bigsqcup \cdots \bigsqcup X_r$ and $Y=Y_1\bigsqcup \cdots \bigsqcup Y_s$, the regular pair of bipartite partitions condition is