  We continue a sequence of recent works studying Ramsey functions for
semialgebraic predicates in $\mathbb{R}^d$. A $k$-ary semialgebraic predicate
$\Phi(x_1,\ldots,x_k)$ on $\mathbb{R}^d$ is a Boolean combination of polynomial
equations and inequalities in the $kd$ coordinates of $k$ points
$x_1,\ldots,x_k\in\mathbb{R}^d$. A sequence $P=(p_1,\ldots,p_n)$ of points in
$\mathbb{R}^d$ is called $\Phi$-homogeneous if either $\Phi(p_{i_1},
\ldots,p_{i_k})$ holds for all choices $1\le i_1 < \cdots < i_k\le n$, or it
holds for no such choice. The Ramsey function $R_\Phi(n)$ is the smallest $N$
such that every point sequence of length $N$ contains a $\Phi$-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 $\Phi$
in dimension $2^{k-4}$ with $R_\Phi$ bounded below by a tower of height $k-1$.
We reduce the dimension in their construction, obtaining a $k$-ary
semialgebraic predicate $\Phi$ on $\mathbb{R}^{k-3}$ with $R_\Phi$ 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
$\Omega(n)$ of height $d$ as a lower bound, matching an upper bound by Suk up
to the constant in front of $n$.

Combinatorics