# Article

Full entry | PDF   (0.2 MB)
Keywords:
structure; line; circle; horocycle
Summary:
The main results are the inequalities (1) and (6) for the minimal number of $(r,q)$-structure classes,which improve the ones from [3], and also some geometrical connections, especially the inequality (13).
References:
[1] Bálint, V.: On a certain class of incidence structures. Práce a štúdie Vysokej Školy Dopravnej v Žiline 2 (1979), 97-106 (In Slovak; summary in English, German and Russian). MR 0675948
[2] Bálint, V., Bálintová, A.: On the number of circles determined by $n$ points in Euclidean plane. Acta Mathematica Hungarica 63 (3-4) (1994), 283-289. MR 1261471
[3] Bálint, V., Lauron, Ph.: Some inequalities for the $(r,q)$-structures. STUDIES OF UNIVERSITY OF TRANSPORT AND COMMUNICATIONS IN ŽILINA, Mathematical - Physical Series, Volume 10 (1995), 3-10. MR 1643894
[4] Beck, J.: On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorical geometry. Combinatorica 3 (3-4) (1983), 281-297. MR 0729781
[5] Borwein, P., Moser, W. O. J.: A survey of Sylvester’s problem and its generalizations. Aequa. Math. 40 (1990), 111-135. MR 1069788
[6] de Bruijn, N. G., Erdős, P.: On a combinatorical problem. Nederl. Acad. Wetensch. 51 (1948), 1277-1279. MR 0028289
[7] Csima, J., Sawyer, E. T.: A short proof that there exist $6n/13$ ordinary points. Discrete and Computational Geometry 9 (1993), no. 2, 187-202. MR 1194036
[8] Elekes, G.: $n$ points in the plane can determine $n^{3\over 2}$ unit circles. Combinatorica 4 (1984), 131. MR 0771719 | Zbl 0561.52009
[9] Elliott, P. D. T. A.: On the number of circles determined by $n$ points. Acta Math. Acad. Sci. Hung. 18 (3-4) (1967), 181-188. MR 0213939 | Zbl 0163.14701
[10] Erdős, P.: Néhány geometriai problémáról. Mat. Lapok 8 (1957), 86-92. MR 0098072
[11] Erdős, P.: On some metric and combinatorical geometric problems. Discrete Math. 60 (1986), 147-153. MR 0852104
[12] Hansen, S.: Contributions to the Sylvester-Gallai-Theory. Doctoral dissertation, University of Copenhagen, 1981.
[13] Harborth, H., Mengersen, I.: Point sets with many unit circles. Discrete Math. 60 (1985), 193-197. MR 0852106
[14] Harborth, H.: Einheitskreise in ebenen Punktmengen. 3.Kolloquium über Diskrete Geometrie, Institut für Mathematik der Universität Salzburg (1985), 163-168. Zbl 0572.52020
[15] Jucovič, E.: Problem $24$. Combinatorical Structures and their Applications, New York-London-Paris, Gordon and Breach, 1970.
[16] Kelly, L. M., Moser, W. O. J.: On the number of ordinary lines determined by $n$ points. Canad. J. Math. 10 (1958), 210-219. MR 0097014
[17] Klee, V., Wagon, S.: Old and New Unsolved Problems in Plane Geometry and Number Theory. Mathematical Assoc. Amer., Washington, DC, 1991. MR 1133201
[18] Sylvester, J. J.: Mathematical Question $11851$. Educational Times 59 (1893), 98.

Partner of