2 problems

Let $f_k(n)$ denote the maximum size of a subset of the $n\times n$ grid containing at most $k$ points on each line. Here $k=k(n)=\omega(1)$ as $n\to\infty$. Kovács–Nagy–Szabó's as…

Let $w'$ be a word of length $k\ge 2$ whose letters are all distinct, and let $f(w',n,2)$ denote the maximum number of copies of $w'$ appearing in an $n\times n$ grid. Distinct-let…