5 problems

Let $A$ be a dense $n$-element point set in general position in the plane, and let $K_n[A]$ denote the complete geometric graph induced by $A$. A path is monotone if its edge set i…

Polynomial bound conjecture. We have

Fix an integer $p\geq 1$. Let $A_k(n;q,p)$ denote the color-avoiding Ramsey number for monotone paths, and let $T_h(x)$ denote a tower function of height $h$. Asymptotic tower form…

Let $P$ be a simple $3$-dimensional polytope with $2n$ vertices, and let $f$ be a generic linear functional on $P$. Let $\b5(P,f)$ denote the number of $f$-monotone paths on $P$, a…

Lavrov–Loh conjecture. With probability tending to $1$ as $n\to\infty$, a random edge-ordering of $K_n$ contains a Hamiltonian monotone path.