  The study of ordered Ramsey numbers of monotone paths for graphs and
hypergraphs has a long history, going back to the celebrated work by Erd\H{o}s
and Szekeres in the early days of Ramsey theory. In this paper we obtain
several results in this area, establishing two conjectures of Mubayi and Suk
and improving bounds due to Balko, Cibulka, Kr\'al and Kyn\v{c}l. We also
obtain a color-monotone version of the well-known Canonical Ramsey Theorem of
Erd\H{o}s and Rado, which could be of independent interest.
