Gir27ao–Janzer–Janzer linear-color exponent conjecture for ordered paths

Let $P^<_{k,n}$ be the ordered graph containing all edges of length at most $k$, and let $R_<(P^<_{k,n};q)$ denote the $q$-color ordered Ramsey number. For all positive integers $k,n,q$, there are constants $C>0$ and $D=D(k,q)>0$ such that

$$R_<(P^<_{k,n};q)\leq Dn^{Cq}.$$

Gir27ao–Janzer–Janzer conjecture. The ordered Ramsey number of $P^<_{k,n}$ in $q$ colors admits the displayed bound with an exponent linear in $q$.

The source contrasts this with the known upper bound $Dn^{Cq\log q}$ and records the conjecture as an open improvement for ordered graphs of bounded bandwidth.