Random multilinear maps and the Erdős box problem

Random multilinear maps and the Erdős box problem, Discrete Analysis 2021:17, 8 pp. A major theme in extremal combinatorics is determining the maximum number of edges that a graph or hypergraph can have if it does not contain a certain fixed graph or hypergraph. In the case of graphs, Turán's theorem gives an exact result when the forbidden subgraph is the complete graph $K_r$ (the maximum number of edges is attained when the graph is an $(r-1)$-partite graph with vertex sets as equal as possible in size), and the Erdős-Stone theorem shows that if the forbidden subgraph has chromatic number $r$ then the same example is at least asymptotically best possible. When $r=2$, meaning that the forbidden subgraph is bipartite, the Erdős-Stone theorem merely tells us that the asymptotic density of the graph is zero, which can be proved by a relatively simple direct argument, first noted by Kővári, Sós and Turán. The _Zarankiewicz problem_, which has led to a large amount of research, is to obtain more precise bounds for the number of edges. Even for general complete bipartite graphs there is a significant gap between the best known upper and lower bounds. One case of interest is that of 4-cycles. If $G$ is a graph with $n$ vertices and we write $N(x)$ for the neighbourhood of a vertex $x$, then the number of edges of $G$ is $\sum_x|N(x)|$ (ignoring a factor 2) and the number of labelled 4-cycles, including degenerate ones, is $\sum_{x,y}|N(x)\cap N(y)|^2$. By Cauchy-Schwarz, the latter quantity is at least $n^{-2}(\sum_{x,y}|N(x)\cap N(y)|)^2$. But $\sum_{x,y}|N(x)\cap N(y)|$ counts the number of labelled and possibly degenerate paths of length 2, so it is equal to $\sum_z|N(z)|^2$, which is up to a constant also the number of degenerate 4-cycles. Thus, there is a non-degenerate cycle provided that $$n^{-2}(\sum_z|N(z)|^2)^2\gg\sum_z|N(z)|^2,$$ or in other words provided that $\sum_z|N(z)|^2\gg n^2$. But by Cauchy-Schwarz again, $\sum_z|N(z)|^2\geq n^{-1}(\sum_z|N(z)|)^2=m^2n^{-1}$. Thus, if $m\gg n^{3/2}$, then $G$ contains a non-degenerate 4-cycle. One might expect an upper bound proved in as natural a way as this to be matched by a lower bound given by the standard technique of picking a random graph and removing a few edges. However, if we choose edges independently with probability $p$, then the expected number of (labelled and possibly degenerate) 4-cycles is $p^4n^4$ while the number of edges is $pn^2$ (up to a constant), so for this to work we need $p^3\gg n