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Generalizing Korchmáros—Mazzocca Arcs

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Abstract

In this paper, we generalize the so-called Korchmáros—Mazzocca arcs, that is, point sets of size q + t intersecting each line in 0, 2 or t points in a finite projective plane of order q. For t ≠ 2, this means that each point of the point set is incident with exactly one line meeting the point set in t points.

In PG(2, pn), we change 2 in the definition above to any integer m and describe all examples when m or t is not divisible by p. We also study mod p variants of these objects, give examples and under some conditions we prove the existence of a nucleus.

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Acknowledgement

The authors are grateful to the anonymous referees for their valuable comments and suggestions which have certainly improved the quality of the manuscript.

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Authors and Affiliations

  1. Department of Geometry, MTA-ELTE Geometric and Algebraic Combinatorics Research Group, ELTE Eötvös Loránd University, Budapest, Hungary

    Bence Csajbók

  2. MTA—ELTE Geometric and Algebraic Combinatorics Research Group, Budapest, Hungary

    Zsuzsa Weiner

  3. Prezi.com, H-1065, Budapest, Nagymező utca 54–56, Hungary

    Zsuzsa Weiner

Authors
  1. Bence Csajbók
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  2. Zsuzsa Weiner
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Corresponding author

Correspondence to Zsuzsa Weiner.

Additional information

We dedicate our work to the memory of our high school mathematics teacher, Dr. János Urbán to whom we are both very grateful.

The first author was partially supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences and by the National Research, Development and Innovation Office — NKFIH, grant no. PD 132463. Both authors acknowledge the support of the National Research, Development and Innovation Office — NKFIH, grant no. K 124950.

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Csajbók, B., Weiner, Z. Generalizing Korchmáros—Mazzocca Arcs. Combinatorica 41, 601–623 (2021). https://doi.org/10.1007/s00493-020-4419-z

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  • DOI: https://doi.org/10.1007/s00493-020-4419-z

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