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

This is the part-preserving homomorphism density of a four-cycle in a bipartite graph with independently uniform labels $x,x^{\prime }$ in the first part and $y,y^{\prime }$ in the second. Repeated labels are included. It equals the fourth power of the four-cycle norm of the adjacency indicator function. It differs at finite sizes from a count requiring four distinct vertices.