5 problems

Let $n\geq 6$ be even, and let $A_n^*$ denote the maximal area among all small $n$-gons. An optimal $n$-gon is a small $n$-gon whose area is $A_n^*$. Its diameter graph is the grap…

Let $c(d,n,k)$ denote the extremal signed-sum quantity studied in the paper, and let $\omega_{d+1}$ be the corresponding extremal vector configuration. For a linear subspace…

Let $d\ge 2$ and $n\ge 2$. Let $f_d(n)$ denote the largest volume of a $d$-dimensional open set with $n$ connected components and no pair of points at an integral distance. Let…

Let $K$ be a convex body in $\mathbb{R}^d$, let $T^d$ be a $d$-dimensional simplex, let $p\geqslant 1$, and let $n\geqslant d+1$. Write $\mathbb{E}^p_*(K)$ and $\mathbb{E}^p_n(K)$…

Near-linear upper-bound conjecture. The function $S_{P}^{\nparallel}(n)$ should satisfy this bound for every parallelogram-free pattern $P$. The paper suggests that the constructio…