Ordered Ramsey numbers of bounded-bandwidth graphs

Determine whether the ordered Ramsey number of P^<_{k,n}, the ordered graph containing all edges of length at most k, is at most c'(k)n^C for an absolute constant C and a constant c'(k)>0.

Background

The survey describes successive improvements for ordered Ramsey numbers of bounded-bandwidth graphs and states that a question about an absolute polynomial exponent was later answered positively. Because the question is explicitly stated as an unresolved problem in the cited work, it qualifies as an explicitly stated open question in the paper's historical account.

References

Gishboliner, Jin, and Sudakov proved that $R_<(P<_{k,n}) \leq c n{4k-2}$ for some constant $c=c(k)>0$ and asked whether in fact $R_<(P<_{k,n}) \leq c' nC$ for a constant $c'=c'(k)>0$ and an absolute constant $C>0$.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Discussion preceding Theorem (thm-gjj24), Section 2.1

Is $R_<(P<_{k,n})= n{2+o(1)}$ as $n \to \infty$?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem in Subsection “Bounded bandwidth”