KöMaL 2017 A698

Let $m$ and $n$ be positive integers, and let $H$ denote a subset of the set $\{1,2,\ldots,m\}\times\{1,2,\ldots,n\}$. Show that if $|H|>m+(m+n)\log_2n$, then there exist integers $1\le u<v\le m$ and $1\le x<y<z\le n$ such that the pairs $(u,x)$, $(u,y)$, $(v,x)$ and $(v,z)$ are elements of $H$.