7 problems

For positive integers $r$, $t$, and $n$, let $P_n^t$ be the $t$-th power of the path on $n$ vertices, and let $R_<^r(P_n^t)$ denote the $r$-colour ordered Ramsey number of $P_n^t$.…

Interval-chromatic-two conjecture. There exists an $\epsilon>0$ such that, for every ordered matching $M^<$ on $n$ vertices with $\chi_<(M^<)=2$,

Let $NM_k$ be the ordered matching on $[2k]$ whose edges are $(i,j)$ with $i+j=2k+1$, and let $r_<(G,H)$ denote the off-diagonal ordered Ramsey number. Nested matching conjecture.…

Let $H$ and $H'$ be $\chi$-unavoidable connected ordered graphs, each with at least two edges. Connected ordered-graph finiteness conjecture. The pair $(H,H')$ is Ramsey finite if a…

Let $H$ and $H'$ be ordered forests, and let $R_<(H,H')$ denote the family of Ramsey graphs of $(H,H')$. Forest Ramsey-graph conjecture. If $(H,H')$ is Ramsey finite, then…

Let $H$ and $H'$ be ordered graphs with asymmetric $2$-density parameters satisfying $m_2(H)\geq m_2(H')$. Let $m(F)$ denote the relevant density of a graph $F$, and let…

Let $H$ and $H'$ be ordered forests, and let $t$ be a positive integer. Write $R_<(H,H')$ for the family of Ramsey graphs of $(H,H')$, and let $\operatorname{girth}(F)$ denote the…