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We call a point sequence P in Rd order-type homogeneous if all (d + 1)-tuples in P have the same orientation. We also provide a natural geometric Ramsey-type theorem with a large Ramsey function. We reduce the dimension in their construction, obtaining a k-ary semialgebraic predicate Φ on Rk−3 with RΦ bounded below by a tower of height k − 1. The Ramsey function RΦ (n) is the smallest N such that every point sequence of length N contains a Φ-homogeneous subsequence of length n.Ĭonlon, 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 ≥ 4, they exhibit a k-ary Φ in dimension 2k−4 with RΦ bounded below by a tower of height k − 1. A sequence P = (p1, …, pn) of points in Rd is called Φ-homogeneous if either Φ(pi1, …,pik) holds for all choices 1 ≤ i1 < … < ik ≤ n, or it holds for no such choice. A k-ary semialgebraic predicate Φ(x1, …, xk) on Rd is a Boolean combination of polynomial equations and inequalities in the kd coordinates of k points x1, …, xk ∈ Rd. We continue a sequence of recent works studying Ramsey functions for semialgebraic predicates in Rd. This lends further support for Sierksma's conjecture on the number of Tverberg partitions. Our techniques also yield a large family of $T(d,r)$-point sets for which the number of Tverberg partitions is exactly $(r-1)!^d$. Along the way, we study the avoidability of many other geometric predicates, and we raise many open problems. We conjecture a complete characterization of the unavoidable Tverberg partitions, and we prove some cases of our conjecture for $d\le 4$. In this paper we study the problem of determining which Tverberg partitions are unavoidable. We say that $\mathcal I$ is "unavoidable" if it occurs in every sufficiently long point sequence. For positive integers $N$ and $r \geq 2$, an $r$-monotone coloring of $\binom$ into $r$ parts "occurs" in an ordered point sequence $P$ if $P$ contains a subsequence $P'$ of $T(d,r)$ points such that the partition of $P'$ that is order-isomorphic to $\mathcal I$ is a Tverberg partition.