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Topological designs. (English) Zbl 1284.57020

Summary: We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves which can be placed on a closed surface of genus \(g\) such that any two of the curves intersects at most once. Although the gap is large, both bounds are the best known for large genus. In genus one and two, we solve the problem exactly. Our methods generalize to variants in which the allowed number of pairwise intersections is odd, even, or bounded, and to surfaces with boundary components.

MSC:

57M99 General low-dimensional topology
05B99 Designs and configurations

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References:

[1] Aougab, T.: Large collections of curves pairwise intersecting exactly once. Preprint arXiv:1210.2797. http://arxiv.org/abs/1210.2797 (2012) · Zbl 1303.57018
[2] Birman, J.S., Hilden, H.M.: On the mapping class groups of closed surfaces as covering spaces. In: Advances in the theory of Riemann surfaces (Proc. Conf., Stony Brook, NY, 1969), pp. 81-115. Annals of Mathematical Studies, No. 66. Princeton University Press, Princeton, NJ (1971)
[3] Brinkmann, G., McKay, B.D.: Fast generation of planar graphs. MATCH Commun. Math. Comput. Chem. 58(2), 323-357 (2007) · Zbl 1164.68025
[4] Farb, B., Margalit, D.: A primer on mapping class groups, Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton, NJ (2012) · Zbl 1245.57002
[5] Haas, A., Susskind, P.: The geometry of the hyperelliptic involution in genus two. Proc. Am. Math. Soc. 105(1), 159-165 (1989). doi:10.2307/2046751 · Zbl 0672.30033 · doi:10.2307/2046751
[6] Juvan, M., Malnič, A., Mohar, B.: Systems of curves on surfaces. J. Combin. Theory Ser. B 68(1), 7-22 (1996). doi:10.1006/jctb.1996.0053 · Zbl 0859.57014 · doi:10.1006/jctb.1996.0053
[7] Stinson, D.R.: Combinatorial designs. Springer, New York. Constructions and analysis, with a foreword by Charles J, Colbourn (2004) · Zbl 1031.05001
[8] Turán, P.: Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48, 436-452 (1941) · Zbl 0026.26903
[9] Vu, V.H.: Extremal set systems with weakly restricted intersections. Combinatorica 19(4), 567-587 (1999). doi:10.1007/s004939970008 · Zbl 0985.05051 · doi:10.1007/s004939970008
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