Above and below

Abstract

We study a family of above-below Ramsey functions $\operatorname{AB}^{(d)}(k)$ defined for sequences of points in $\mathbb R^d$ whose projections to $\mathbb R^{d-1}$ have cyclic order type. The case $d=3$ is the above-below function $\operatorname{AB}(k)$ that was first introduced by Pohoata and Zakharov in their work on the Erdős-Szekeres problem in $\mathbb{R}^{3}$. We prove the sharp estimate \[ \operatorname{AB}(k)=2^{2^{Θ(k)}}, \] and, more generally, show that $\operatorname{AB}^{(d)}(k)$ is closely related to the higher-order cup-cap function of Eliáš and Matoušek and the monotone Ramsey numbers of Balko.