Ramsey numbers and monotone colorings

For positive integers $N$ and $r\ge 2$, an $r$-monotone coloring of $\left(\genfrac{}{}{0px}{}{\{1,\dots ,N\}}{r}\right)$ is a 2-coloring by $-1$ and $+1$ that is monotone on the lexicographically ordered sequence of $r$-tuples of every $(r+1)$-tuple from~$\left(\genfrac{}{}{0px}{}{\{1,\dots ,N\}}{r+1}\right)$. Let ${\bar{R}}_{mon}(n;r)$ be the minimum $N$ such that every $r$-monotone coloring of $\left(\genfrac{}{}{0px}{}{\{1,\dots ,N\}}{r}\right)$ contains a monochromatic copy of $\left(\genfrac{}{}{0px}{}{\{1,\dots ,n\}}{r}\right)$. For every $r\ge 3$, it is known that ${\overline{R}{mon}}(n;r) \leq tow{r-1}(O(n))$,where$tow_h(x)$isthetowerfunctionofheight$h-1$definedas$tow_1(x)=x$and$tow_h(x) = 2^{tow_{h-1}(x)}$for$h \geq 2$. The Erd\H{o}s–Szekeres Lemma and the Erd\H{o}s–Szekeres Theorem imply ${\bar{R}}_{mon}(n;2)=(n-1{)}^2+1$ and ${\bar{R}}_{mon}(n;3)=\left(\genfrac{}{}{0px}{}{2n-4}{n-2}\right)+1$, respectively. It follows from a result of Eli'{a}\v{s} and Matou\v{s}ek that ${\overline{R}_{mon}}(n;4)\geq tow_3(\Omega(n))$. We show that ${\bar{R}}_{mon}(n;r)\ge to{w}_{r-1}(\mathrm{\Omega }(n))$ for every $r\ge 3$. This, in particular, solves an open problem posed by Eli'{a}\v{s} and Matou\v{s}ek and by Moshkovitz and Shapira. Using two geometric interpretations of monotone colorings, we show connections between estimating ${\bar{R}}_{mon}(n;r)$ and two Ramsey-type problems that have been recently considered by several researchers. Namely, we show connections with higher-order Erd\H{o}s–Szekeres theorems and with Ramsey-type problems for order-type homogeneous sequences of points. We also prove that the number of $r$-monotone colorings of $\left(\genfrac{}{}{0px}{}{\{1,\dots ,N\}}{r}\right)$ is $2^{N^{r-1}\mathrm{/}{r}^{\mathrm{\Theta }(r)}}$ for $N \geq r \geq 3$, which generalizes the well-known fact that the number of simple arrangements of~$N$ pseudolines is $2^{\mathrm{\Theta }(N^2)}$.