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How do 9 Points Look Like in \(\mathbb {E}^3\)?


The aim of this note is to give an elementary proof of the following fact: given three red convex sets and three blue convex sets in \(\mathrm{I\!E}^3\), such that every red intersects every blue, there is a line transversal to the reds or there is a line transversal to the blues. This is a special case of a theorem of Montajano and Karasev (Discrete Comput Geom 46(2):283–300, 2011) and generalizes, in a sense, the colourful Helly theorem due to Lovász (cf. Discrete Math 40(2,3):141–152, 1982)

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  1. Arocha, Bracho, Montejano, Strausz; Separoids, Their Categories and a Hadwiger-Type Theorem for Transversals; Discrete & Computational Geometry, 27(3), 377–385 (2002).

  2. Bárány, I; A generalization of Carathéodory’s theorem. Discrete Math. 40(2,3), 141–152 (1982).

  3. Karasev, Montejano; Topological transversals to a family of convex sets; Discrete & Computational Geometry, 46(2), 283–300 (2011).

  4. Montejano; Transversals, Topology and Colorful Geometric Results; Geometry–Intuitive, Discrete, and Convex; Bolyai Society Mathematica Studies 24, 1–14.

  5. Strausz; Erdős-Szekeres “happy end”-type theorems for separoïds; European Journal of Combinatorics 29(4), 1076–1085 (2008).

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Correspondence to Ricardo Strausz.

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Communicated by Victor Chepoi.

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Strausz, R. How do 9 Points Look Like in \(\mathbb {E}^3\)?. Ann. Comb. 26, 303–307 (2022).

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