Geodesic
Partitioning of plane sets into $6$ subsets of small diameter
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XII, Tome 497 (2020), pp. 100-123 Cet article a éte moissonné depuis la source Math-Net.Ru

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In 1956, H. Lenz introduced a problem, which was to find members of the sequence $$d_n=\inf_{\Phi}\{x\in \mathbb{R}^{+}:\Phi \subset \Phi_1 \cup \Phi_2 \cup \dots \cup \Phi_n, \forall i \,\textrm{diam}\, \Phi_i \leq x \}$$ where infimum is taken over all sets $\Phi$ of unit diameter. In this paper, we improve an upper bound for $d_6$ to $0.53432\dots $. 
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V. O. Koval'. Partitioning of plane sets into $6$ subsets of small diameter. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part XII, Tome 497 (2020), pp. 100-123. http://geodesic.mathdoc.fr/item/ZNSL_2020_497_a4/

[1] K. Borsuk, “Drei Sátze über die $n$-dimensional euklidische Sphäre”, Fundamenta Math., 20 (1933), 177–190 | DOI

[2] A. M. Raigorodskii, “Coloring Distance Graphs and Graphs of Diameters”, Thirty Essays on Geometric Graph Theory, ed. J. Pach, Springer, 2013, 429–460 | DOI | MR | Zbl

[3] A. M. Raigorodskii, “Cliques and cycles in distance graphs and graphs of diameters”, Discrete Geometry and Algebraic Combinatorics, Contemporary Mathematics, 625, AMS, 2014, 93–109 | DOI | MR | Zbl

[4] A. M. Raigorodskii, “Vokrug gipotezy Borsuka”, Itogi nauki i tekhniki. Seriya “Sovremennaya matematika”, 23, 2007, 147–164

[5] V. G. Boltyanski, H. Martini, P. S. Soltan, Excursions into combinatorial geometry, Universitext, Springer, Berlin, 1997 | DOI | MR | Zbl

[6] P. Brass, W. Moser, J. Pach, Research problems in discrete geometry, Springer, 2005 | MR | Zbl

[7] H. Lenz, “Zerlegung ebener Bereiche in konvexe Zellen von möglichst kleinem Durchmesser”, Jber. Deutch. Math. Verein., 58 (1956), 87–97 | MR | Zbl

[8] H. Lenz, “Uber die Bedeckung ebenem Punktmengen durch solche kleineren Durchmessers”, Arch. Math. (Basel), 7:1 (1956), 34–40 | DOI | MR | Zbl

[9] V. P. Filimonov, “O pokrytii ploskikh mnozhestv”, Matematicheskii sbornik, 201:8 (2010), 127–160 | Zbl

[10] D. Belov, N. Aleksandrov, “O razbienii ploskikh mnozhestv na shest chastei malogo diametra”, TRUDY MFTI, 4:1 (2012), 77

[11] Proverka raschetov, https://github.com/VadymKoval/calculations-check.git

[12] V. P. Filimonov, “O pokrytii mnozhestv v $\mathbb{R}^m$”, Matem. sb., 205:8 (2014), 95–138 | MR | Zbl

[13] R. I. Prosanov, “Kontrprimery k gipoteze Borsuka, imeyuschie bolshoi obkhvat”, Matem. zametki, 105:6 (2019), 890–898 | MR | Zbl

[14] A. Kupavskii, A. Polyanskii, “Proof of Schur's conjecture in $\mathbb{R}^d$”, Combinatorica, 37:6 (2017), 1181–1205 | DOI | MR | Zbl

[15] A. Kupavskii, A. Polyanskii, “On simplices in diameter graphs in $\mathbb{R}^4$”, Mathematical Notes, 101:2 (2017), 232–246 | MR | Zbl

[16] A. M. Raigorodskii, L. I. Bogolyubskii, “Ob otsenkakh v probleme Borsuka”, Trudy MFTI, 11:3 (2019), 20–49