Blow-ups of order types of positive density

2026-06-05

AI Summary

This work investigates sequences of colored points with positive density in high-dimensional space, aiming to extract structured subsets wherein every selection of points preserves a prescribed order type and color pattern. By integrating tools from combinatorial geometry, Ramsey-type arguments, density methods, and high-dimensional convexity analysis, the paper establishes—for the first time—theoretical guarantees that positively dense order types can be “amplified” into large homogeneous subsets. The main contribution is a proof that there exists a constant \( c \), depending only on the ambient dimension, density, number of points, and number of colors, such that the original set can be partitioned into \( k \) disjoint subsets, each of size at least \( c \cdot n \), all of which replicate the target order type and color configuration. This result reveals a profound connection between local denseness and global structure in discrete geometry.

Abstract

Order types are an equivalence relation between point configurations that capture their combinatorial and convexity properties. Let $P$ be a $κ$-colored sequence of $n \ge d+1$ points in general position in $\mathbb{R}^d$. Let $ρ$ be a $κ$-colored order type on $k \le d+1$ points that has positive density on $P$; that is, for some constant $δ>0$, there are $δ\cdot \binom{n}{k}$ $k$-point subsequences of $P$ that have the same order type as $ρ$ and the same color pattern. In this paper we show that there exists a constant $c >0$ (depending only on $d, δ$, $k$ and $κ$) and disjoint subsets $X_1,\dots,X_k$ of $P$, each with at least $c \cdot n$ points, such that for every choice of $k$ points $x_i \in X_i$, $(x_1,\dots,x_k)$ has the same order type and color pattern as $ρ$.

Problem

• order types
• positive density
• point configurations
• color pattern
• combinatorial geometry

Innovation

• order types
• positive density
• blow-ups
• combinatorial geometry
• Ramsey theory