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Blow-ups of order types of positive density
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 ≥ d+1 points in general position in ℝd. Let ρ be a κ-colored order type on k ≤ d+1 points that has positive density on P; that is, for some constant δ>0, there are δ· binomnk 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 X1,…,Xk of P, each with at least c · n points, such that for every choice of k points xi ∈ Xi, (x1,…,xk) has the same order type and color pattern as ρ.
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BibTeX
@misc{fabilamonroy2026blowups,
title = {{Blow-ups of order types of positive density}},
author = {Ruy Fabila-Monroy and Benedikt Hahn and Jesús Leaños},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.07806},
eprint = {2606.07806},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.07806},
}
RIS
TY - UNPB
TI - Blow-ups of order types of positive density
AU - Ruy Fabila-Monroy
AU - Benedikt Hahn
AU - Jesús Leaños
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2606.07806
UR - https://arxiv.org/abs/2606.07806
AB - 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 ≥ d+1 points in general position in ℝ^d. Let ρ be a κ-colored order type on k ≤ d+1 points that has positive density on P; that is, for some constant δ>0, there are δ· binomnk 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,…,X_k of P, each with at least c · n points, such that for every choice of k points x_i ∈ X_i, (x_1,…,x_k) has the same order type and color pattern as ρ.
ER -