Asymptotics for bounded-bandwidth ordered graphs

Determine whether, for every fixed bandwidth k, the ordered Ramsey number of the graph P^<_{k,n} containing all edges of length at most k satisfies R_<(P^<_{k,n}) ≤ c' n^C for constants c'=c'(k)>0 and an absolute constant C>0.

Background

The survey discusses successive improvements for ordered Ramsey numbers of graphs whose edge lengths are at most k. The cited question asks whether the exponent in the polynomial upper bound can be made absolute, independent of k, although later work in the survey reports a positive answer.

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 Section 2, subsection “Bounded bandwidth”