Lower bounds on geometric Ramsey functions

We continue a sequence of recent works studying Ramsey functions for semialgebraic predicates in ${\mathbb{R}}^d$. A $k$-ary semialgebraic predicate $\mathrm{\Phi }(x_1,\dots ,x_k)$ on ${\mathbb{R}}^d$ is a Boolean combination of polynomial equations and inequalities in the $kd$ coordinates of $k$ points $x_1,\dots ,x_k\in {\mathbb{R}}^d$. A sequence $P=(p_1,\dots ,p_n)$ of points in ${\mathbb{R}}^d$ is called $\mathrm{\Phi }$-homogeneous if either $\Phi(p_{i_1}, \ldots,p_{i_k})$holdsforallchoices$1\le i_1 < \cdots < i_k\le n$, or it holds for no such choice. The Ramsey function $R_{\mathrm{\Phi }}(n)$ is the smallest $N$ such that every point sequence of length $N$ contains a $\mathrm{\Phi }$-homogeneous subsequence of length $n$. Conlon, Fox, Pach, Sudakov, and Suk constructed the first examples of semialgebraic predicates with the Ramsey function bounded from below by a tower function of arbitrary height: for every $k\ge 4$, they exhibit a $k$-ary $\mathrm{\Phi }$ in dimension $2^{k-4}$ with $R_{\mathrm{\Phi }}$ bounded below by a tower of height $k-1$. We reduce the dimension in their construction, obtaining a $k$-ary semialgebraic predicate $\mathrm{\Phi }$ on ${\mathbb{R}}^{k-3}$ with $R_{\mathrm{\Phi }}$ bounded below by a tower of height $k-1$. We also provide a natural geometric Ramsey-type theorem with a large Ramsey function. We call a point sequence $P$ in ${\mathbb{R}}^d$ order-type homogeneous if all $(d+1)$-tuples in $P$ have the same orientation. Every sufficiently long point sequence in general position in ${\mathbb{R}}^d$ contains an order-type homogeneous subsequence of length $n$, and the corresponding Ramsey function has recently been studied in several papers. Together with a recent work of B'ar'any, Matou\v{s}ek, and P'or, our results imply a tower function of $\mathrm{\Omega }(n)$ of height $d$ as a lower bound, matching an upper bound by Suk up to the constant in front of $n$.