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  "path": "/abs/2606.07806v1",
  "publishedAt": "2026-06-09T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Ruy Fabila-Monroy",
    "Benedikt Hahn",
    "Jesús Leaños"
  ],
  "textContent": "**Authors:** Ruy Fabila-Monroy, Benedikt Hahn, Jesús Leaños\n\nOrder 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 $ρ$.",
  "title": "Blow-ups of order types of positive density"
}