Multicolor bounded-bandwidth ordered paths

Prove that, 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)≤Dn^{Cq}.

Background

A bound of the form Dn{Cq log q} is known for multicolor ordered paths of bandwidth k. The conjecture asks whether the logarithmic factor in q can be removed.

References

They also mention some interesting consequences for the problem of finding monochromatic directed paths in multicolored tournaments and conjecture that their upper bound can be improved.

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

They also mention some interesting consequences for the problem of finding monochromatic directed paths in multicolored tournaments and conjecture that their upper bound can be improved.

For all positive integers $k,n,q$, there are constants $C>0$ and $D=D(k,q)>0$ such that $R_<(P<_{k,n};q) \leq Dn{Cq}$.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Conjecture following Theorem (unnumbered), Section 2.5

For all positive integers $k,n,q$, there are constants $C>0$ and $D=D(k,q)>0$ such that $R_<(P<_{k,n};q) \leq Dn{Cq}$.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Conjecture in subsection “Multicolor Ordered Ramsey Numbers”