The study of ordered Ramsey numbers of monotone paths for graphs and hypergraphs has a long history, going back to the celebrated work by Erdős 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, Král and Kynčl. For example, in the graph case, we show that the ordered Ramsey number for a fixed clique versus a fixed power of a monotone path of length n is always linear in n. Also, in the 3-graph case, we show that the ordered Ramsey number for a fixed clique versus a tight monotone path of length n is always polynomial in n. As a by-product, we also obtain a color-monotone version of the well-known Canonical Ramsey Theorem of Erdős and Rado, which could be of independent interest.

图和超图的单调路径有序拉姆齐数的研究有着悠久的历史，可以追溯到拉姆齐理论早期 Erdős 和 Szekeres 的著名著作。在本文中，我们在该领域取得了一些成果，建立了 Mubayi 和 Suk 的两个猜想，并改进了 Balko、Cibulka、Král 和 Kynčl 的界限。例如，在图的情况下，我们表明固定团的有序拉姆齐数与长度为n的单调路径的固定幂在n中始终是线性的。此外，在 3 图情况下，我们表明固定团与长度为n的紧单调路径的有序拉姆齐数始终是n中的多项式。作为副产品，我们还获得了著名的 Erdős 和 Rado 的正则拉姆齐定理的颜色单调版本，这可能具有独立意义。