Upper bounds for bounded-degree ordered 3-uniform hypergraphs

Determine an upper bound on ordered Ramsey numbers of ordered 3-uniform hypergraphs with bounded maximum degree.

Background

The survey explicitly states that no nontrivial upper bounds are known in general for ordered 3-uniform hypergraphs with bounded maximum degree. The problem asks for any meaningful general upper estimate.

References

In general, we are not aware of any nontrivial upper bounds on ordered Ramsey numbers of ordered $3$-uniform hypergraphs with bounded maximum degree.

What is the upper bound on ordered Ramsey numbers of ordered $3$-uniform hypergraphs with bounded maximum degree?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Problem following the discussion of bounded-degree ordered hypergraphs, Section 3.3

What is the upper bound on ordered Ramsey numbers of ordered $3$-uniform hypergraphs with bounded maximum degree?

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Section 3, subsection “Bounded degrees and interval chromatic number”