Above and below

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 Erd\H{o}s-Szekeres problem in $\mathbb{R}^{3}$. We prove the sharp estimate $$\operatorname{AB}(k)=2^{2^{\Theta(k)}},$$ and, more generally, show that $\operatorname{AB}^{(d)}(k)$ is closely related to the higher-order cup-cap function of Eli\'a\v{s} and Matou\v{s}ek and the monotone Ramsey numbers of Balko.