10 problems

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 $h$. The parameters $k$, $n$, $q$, and $p$ are posit…

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 an $n$-dimensional polytope and let $\ell$ be a generic linear function on $P$. Let $G(P,\ell)$ be the graph whose vertices are the $\ell$-monotone paths connecting the…

For positive integers $n\geq k$ and $r>s$, let $A_k(n;r,s)$ be the least $N$ such that every $r$-coloring of $[N]^{(k)}$ contains a tight monotone path on $n$ edges using at most…

Colour the edges of the complete graph on vertex set $\{1,2,\dots,n\}$ arbitrarily, and let $f(l,k)$ be the smallest integer such that whenever $n>f(l,k)$, there is a monotone mono…

Polynomial bound conjecture. We have

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…

For an edge-ordered complete graph on $n$ vertices, consider the maximum length of a monotone increasing path that must appear. Chvátal–Komlós conjecture. This quantity is linear i…

Chvátal–Komlós altitude conjecture. The parameter $\alpha(K_n)$ has order of magnitude $n$; equivalently, $\alpha(K_n)=\Theta(n)$.

Growth conjecture for $f(K_n)$. The lower bound