Congruent triangles in arrangements of lines. (English) Zbl 1396.52022

In the paper under review, the author studies the maximal number of congruent triangles determined by finite arrangements of \(\ell\) lines in \(\mathbb{R}^{2}\). In order to formulate the main results of the paper, we need to recall some notions. An arrangement \(\mathcal{A}\) of lines is always a finite family of \(\ell\) lines \(L_{1},\dots,L_{\ell}\), which is not a pencil. Denote by \(\mathfrak{U}_{\ell}\) the set of all arrangements of \(\ell \geq 3\) lines. For an arrangement \(\mathcal{A} \in \mathfrak{U}_{\ell}\) we can associate a graph \(\Gamma_{\mathcal{A}}\): the vertices of \(\Gamma_{\mathcal{A}}\) correspond to the intersection points of lines and the edges of \(\Gamma_{\mathcal{A}}\) correspond to the line-segments between these vertices. A triangle in \(\mathcal{A} \in \mathfrak{U}_{\ell}\) is the convex hull of the set of intersection points of three non-concurrent pairwise non-parallel lines in \(\mathcal{A}\). We denote by \(F^{\mathcal{A}}\) the set of all triangles in \(\mathcal{A}\). If two triangles \(\triangle_{1}, \triangle_{2} \in F^{\mathcal{A}}\) are congruent, then we write \(\triangle_{1} \sim \triangle_{2}\). Let \(F_{1}^{\mathcal{A}}, \dots, F_{p}^{\mathcal{A}}\) be the equivalence classes with respect to \(\sim\) such that \(\# F_{1}^{\mathcal{A}} \geq \dots \geq \# F_{p}^{\mathcal{A}}\). We call a triangle \(\triangle \in F^{\mathcal{A}}\) facial if it is a face of \(\Gamma_{\mathcal{A}}\), i.e., \(L \cap \text{int} \triangle = \emptyset\) for all \(L \in \mathcal{A}\). Denote by \(G^{\mathcal{A}} \subset F^{\mathcal{A}}\) the set of all facial triangles, and we denote by \(G_{1}^{\mathcal{A}}, \dots ,G_{q}^{\mathcal{A}}\) the equivalence classes with respect to \(\sim\) such that \(\# G_{1}^{\mathcal{A}} \geq \dots \geq \# G_{q}^{\mathcal{A}}\). Now we define \[ f(\ell) = \max_{\mathcal{A} \in \mathfrak{U}_{\ell}} \# F_{1}^{\mathcal{A}}, \quad \quad g(\ell) = \max_{\mathcal{A} \in \mathfrak{U}_{\ell}} \# G_{1}^{\mathcal{A}}. \] Additionally, we define \(F(\ell)\) (\(G(\ell)\), respectively) as the set of all integers \(u\) such that there exists an arrangement on \(\ell\) lines having exactly \(u\) congruent triangles (congruent facial triangles, respectively). We write \([s..t]\) for the set of all integers \(u\) such that \(s \leq u\leq t\), put \(H\) for \(F\) or \(G\), and \(h\) is equal to \(f\) if \(H=G\) or \(g\) if \(H = G\). Whenever \(H(\ell) = [0..h(\ell)]\), we say that \(H(\ell)\) is complete.
Theorem 1. One has \(f(5) = g(5) = 5\) while \(F(5)\) and \(G(5)\) are complete.
Thereom 2. One has \(f(6) = 8, \, 6 \leq g(6) \leq 7\), \(F(6)\) is complete, and \([0..6] \subset G(6)\).
Theorem 3. One has \(f(7) = 14, \, 9 \leq g(7) \leq 11, [0..10]\cup\{14\} \subset F(7)\), and \([0..9] \subset G(7)\).
Theorem 4. One has \(16 \leq f(8) \leq 22, \, 12 \leq g(8) \leq 15\), \([0..16] \setminus \{13\} \subset F(8)\) and \([0..12] \subset G(8)\).
In the last section, the author formulates some natural conjectures, for instance the author predicts that \(g(6)=6\).


52C10 Erdős problems and related topics of discrete geometry
52C30 Planar arrangements of lines and pseudolines (aspects of discrete geometry)


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[1] [2] T. Christ, Database of Combinatorially Different Line Arrangements,http://www.inf. ethz.ch/personal/christt/line_arrangements.php(accessed on 15 June 2016).
[2] [3] T. Christ, Discrete Descriptions of Geometric Objects, Ph.D. thesis, ETH Z¨urich, 2011, doi: 10.3929/ethz-a-007018092.
[3] [4] P. Erd˝os and G. Purdy, Some extremal problems in geometry III, in: F. Hoffman (ed.), Proceedings of the 6th Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic University, Boca Raton, Florida), Utilitas Mathematica, 1975 pp. 291-308. · Zbl 0328.05018
[4] [5] P. Erd˝os and G. Purdy, Some extremal problems in geometry IV, in: F. Hoffman (ed.), Proceedings of the 7th Southeastern Conference on Combinatorics, Graph Theory, and Computing (Louisiana State University, Baton Rouge, Louisiana), Utilitas Mathematica, 1976 pp. 307- 322. · Zbl 0345.52007
[5] [6] P. Erd˝os and G. Purdy, Extremal problems in combinatorial geometry, in: R. L. Graham, M. Gr¨otschel and L. Lov´asz (eds.), Handbook of Combinatorics, The MIT Press, Cambridge, Massachusetts, volume 1, 1995 pp. 809-874. · Zbl 0852.52009
[6] [7] D. Forge and J. L. Ram´ırez Alfons´ın, Straight line arrangements in the real projective plane, Discrete Comput. Geom.20 (1998), 155-161, doi:10.1007/pl00009373. · Zbl 0921.52004
[7] [14] N. J. A. Sloane, Sequence A006066 (formerly M1334) in The On-Line Encyclopedia of Integer Sequences, published electronically athttps://oeis.org.
[8] [15] E. W. Weisstein, Kobon Triangle, from MathWorld—A Wolfram Web Resource,http:// mathworld.wolfram.
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