6 problems

Let $n$ be sufficiently large, and consider subsets of the $n$-by-$n$ grid with no three collinear points. Guy and Kelly's constant is … with $\epsilon>0$. Guy–Kelly conjecture. Ev…

Monochromatic NTIL asymptotics. The exact monochromatic no-three-in-line maxima satisfy

Let $S_n$ be the set of $n^2$ points in $\mathbb{R}^2$ with integer coordinates $(x,y)$ satisfying $1\leq x,y<n$, and let $f_n$ be the maximum size of a subset of $S_n$ containing…

Let $S_n$ be the set of $n^2$ points in $\mathbb{R}^2$ with integer coordinates $(x,y)$ satisfying $1\leq x,y<n$, and let $f_n$ be the maximum size of a subset of $S_n$ containing…

Let $S_{\mathrm{lex<}}$ and $S_{\mathrm{mod2lex<}}$ be the lexicographic greedy point sets described above, consisting of triple-wise non-collinear integer points constructed with…

Large-grid unattainability conjecture. For sufficiently large $n$, the upper bound $2n$ is unattainable.