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Extremal problems for convex geometric hypergraphs and ordered hypergraphs

Part of: Extremal combinatorics Discrete geometry

Published online by Cambridge University Press:  10 August 2020

Zoltán Füredi ,
Tao Jiang ,
Alexandr Kostochka ,
Dhruv Mubayi  and
Jacques Verstraëte
Zoltán Füredi
Affiliation:
Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca 13–15, H-1053, Budapest, Hungary e-mail: zfuredi@gmail.com
Tao Jiang
Affiliation:
Department of Mathematics, Miami University, Oxford, OH45056, USA e-mail: jiangt@miamioh.edu
Alexandr Kostochka
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL61801, USA and Sobolev Institute of Mathematics, Novosibirsk630090, Russia e-mail: kostochk@math.uiuc.edu
Dhruv Mubayi*
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL60607, USA
Jacques Verstraëte
Affiliation:
Department of Mathematics, University of California at San Diego, 9500 Gilman Drive, La Jolla, CA92093-0112, USA e-mail: jverstra@math.ucsd.edu
*
e-mail: mubayi@uic.edu
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Abstract

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and determine the order of magnitude of the extremal function for various ordered and convex geometric paths and matchings. Our results generalize earlier works of Braß–Károlyi–Valtr, Capoyleas–Pach, and Aronov–Dujmovič–Morin–Ooms-da Silveira. We also provide a new variation of the Erdős-Ko-Rado theorem in the ordered setting.

Keywords

Convex geometric hypergraphextremal hypergraph

MSC classification

Primary: 05D05: Extremal set theory
Secondary: 52C10: Erdős problems and related topics of discrete geometry
Type
Article
Information
Canadian Journal of Mathematics , Volume 73 , Issue 6 , December 2021 , pp. 1648 - 1666
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Copyright
© Canadian Mathematical Society 2020

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Footnotes

Research supported by grant K116769 from the National Research, Development and Innovation Office NKFIH and by the Simons Foundation Collaboration grant #317487. Research partially supported by NSF grants DMS-1400249, DMS-1855542, DMS-1300138, DMS-1763317, and DMS-1556524, as well as DMS-1600592 by Award RB17164 of the UIUC Campus Research Board and by grants 18-01-00353A and 19-01-00682 of the Russian Foundation for Basic Research.

References

Aronov, B., Dujmovič, V., Morin, P., Ooms, A., and da Silveira, L., More Turán-type theorems for triangles in convex point sets . Electron. J. Comb. 26(2019), no. 1, Article Number P1.8.Google Scholar
Braß, P., Turán-type extremal problems for convex geometric hypergraphs . Contemp. Math. 342(2004), 25–34.CrossRefGoogle Scholar
Braß, P., Károlyi, G., and Valtr, P., A Turán-type extremal theory of convex geometric graphs . In: Goodman-Pollack festschrift, Springer-Verlag, Berlin, Turnhout, 2003, pp. 277–302.Google Scholar
Braß, P., Rote, G., and Swanepoel, K., Triangles of extremal area or perimeter in a finite planar point set . Discrete Comput. Geom. 26(2001), 51–58.CrossRefGoogle Scholar
Capoyleas, V. and Pach, J., A Turán-type theorem for chords of a convex polygon . J. Combin. Theory Ser. B 56(1992), 9–15.CrossRefGoogle Scholar
Erdős, P. and Hanani, H., On a limit theorem in combinatorial analysis . Publ. Math. Debr. 10(1963), 10–13.Google Scholar
Erdős, P. and Kleitman, D., On coloring graphs to maximize the proportion of multicolored k-edges . J. Combin. Theory 5(1968), 164–169.CrossRefGoogle Scholar
Füredi, Z., Jiang, T., Kostochka, A., Mubayi, D., and Verstraete, J., Tight paths in convex geometric hypergraphs . Adv. Combinatorics 1(2020), no. 1, 14.Google Scholar
Füredi, Z., Kostochka, A., Mubayi, D., and Verstraete, J., Ordered and convex geometric trees with linear extremal function. Preprint, 2019. https://arxiv.org/abs/1812.05750 CrossRefGoogle Scholar
Füredi, Z., Jiang, T., Kostochka, A., Mubayi, D., and Verstraëte, J., Partitioning ordered hypergraphs. J. Combin. Theory Ser. A 177 (2021),A 105300.CrossRefGoogle Scholar
Hopf, H. and Pannwitz, E., Aufgabe Nr. 167 . Jahresbericht d. Deutsch Math. Verein. 43(1934), 114.Google Scholar
Korándi, D., Tardos, G., Tomon, I., and Weidert, C., On the Turán number of ordered forests . Electron Notes Discret. Math. 61(2017), 773–779.CrossRefGoogle Scholar
Kupitz, Y. S. and Perles, M., Extremal theory for convex matchings in convex geometric graphs . Discrete Comput. Geom. 15(1996), 195–220.CrossRefGoogle Scholar
Marcus, A. and Tardos, G., Excluded permutation matrices and the Stanley-Wilf conjecture . J. Combinatorial Theory Ser. A 107(2004), 153–160.CrossRefGoogle Scholar
Pach, J. and Tardos, G., Forbidden paths and cycles in ordered graphs and matrices . Israel J. Math. 155(2006), 359–380.CrossRefGoogle Scholar
Pach, J. and Pinchasi, R., How many unit equilateral triangles can be generated by $~n~$ points in general position? Am. Math. Mon. 110(2003), 100–106.CrossRefGoogle Scholar
Sutherland, J. W., Lösung der Aufgabe 167 . Jahresbericht Deutsch. Math. Verein. 45(1935), 33–35.Google Scholar