Bipartite four-cycle count

If a bipartite graph with parts $X,Y$ has at least $\delta |X||Y|$ edges, then the number of ordered tuples $(x_1,x_2,y_1,y_2)\in X^2\times Y^2$ for which every $x_i{y}_j$ is an edge is at leastIndeed, if $d(x_1,x_2)$ is the number of common neighbours of $x_1,x_2$ in $Y$, two applications of the Cauchy-Schwarz inequality give