Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 62: Issue 1
A Note on the Erdős Matching Conjecture
Authors:
Ryan R. Martin Iowa State University, Ames, Iowa, USA

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Balázs Patkós HUN-REN Alfréd Rényi Institute of Mathematics, Budapest, Hungary

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Pages:
63–69
Online Publication Date:
25 Mar 2025
Publication Date:
09 Apr 2025
Article Category:
Research Article
DOI:
https://doi.org/10.1556/012.2025.04330
Keywords:
Extremal set theory; matchings; stability
Open access

The Erdős Matching Conjecture states that the maximum size 𝑓 (𝑛, 𝑘, 𝑠) of a family F⊆nkthat does not contain 𝑠 pairwise disjoint sets is max. Ak,s,Bn,k,s, where Ak,s=sk−1k and Bn,k,s=B∈nk:B∩s−1≠∅. The case 𝑠 = 2 is simply the Erdős-Ko-Rado theorem on intersecting families and is well understood. The case 𝑛 = 𝑠𝑘 was settled by Kleitman and the uniqueness of the extremal construction was obtained by Frankl. Most results in this area show that if 𝑘, 𝑠 are fixed and 𝑛 is large enough, then the conjecture holds true. Exceptions are due to Frankl who proved the conjecture and considered variants for 𝑛 ∈ [𝑠𝑘, 𝑠𝑘 + 𝑐𝑠,𝑘 ] if 𝑠 is large enough compared to 𝑘. A recent manuscript by Guo and Lu considers non-trivial families with matching number at most 𝑠 in a similar range of parameters.

In this short note, we are concerned with the case 𝑠 ≥ 3 fixed, 𝑘 tending to infinity and 𝑛 ∈ {𝑠𝑘, 𝑠𝑘 + 1}. For 𝑛 = 𝑠𝑘, we show the stability of the unique extremal construction of size sk−1k=s−1sskk with respect to minimal degree. As a consequence we derive limk→∞fsk+1,k,ssk+1k<s−1s−εsfor some positive constant 𝜀𝑠 which depends only on 𝑠.

  • [1]

    B. Bollobás, D. E. Daykin, and P. Erdős. Sets of independent edges of a hypergraph. Quart. J. Math. Oxford Ser. (2), 27(105):25–32, 1976.

    • Search Google Scholar
    • Export Citation
  • [2]

    S. Das, W. Gan, and B. Sudakov. The minimum number of disjoint pairs in set systems and related problems. Combinatorica, 36(6):623–660, 2016.

    • Search Google Scholar
    • Export Citation
  • [3]

    S. Das and T. Tran. Removal and stability for Erdős-Ko-Rado. SIAM J. Discrete Math., 30(2):1102–1114, 2016.

  • [4]

    P. Erdős. A problem on independent 𝑟-tuples. Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 8:93–95, 1965.

  • [5]

    P. Erdős, Chao Ko, and R. Rado. Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. (2), 12:313–320, 1961.

  • [6]

    P. Frankl. Erdős-Ko-Rado theorem with conditions on the maximal degree. J. Combin. Theory Ser. A, 46(2):252–263, 1987.

  • [7]

    P. Frankl. Extremal set systems. In Handbook of combinatorics, Vol. 1, 2, pages 1293–1329. Elsevier Sci. B. V., Amsterdam, 1995.

  • [8]

    P. Frankl. Improved bounds for Erdős’ matching conjecture. J. Combin. Theory Ser. A, 120(5):1068–1072, 2013.

  • [9]

    P. Frankl. On the maximum number of edges in a hypergraph with given matching number. Discrete Appl. Math., 216(3):562–581, 2017.

  • [10]

    P. Frankl. Proof of the Erdős matching conjecture in a new range. Israel J. Math., 222(1):421–430, 2017.

  • [11]

    P. Frankl. Maximum degree and diversity in intersecting hypergraphs. J. Combin. Theory Ser. B, 144:81–94, 2020.

  • [12]

    P. Frankl. On non-trivial families without a perfect matching. European J. Combin., 84:article no. 103044, 2020.

  • [13]

    P. Frankl and A. Kupavskii. Diversity. J. Combin. Theory Ser. A, 182:article no. 105468, 2021.

  • [14]

    P. Frankl and A. Kupavskii. The Erdős matching conjecture and concentration inequalities. J. Combin. Theory Ser. B, 157:366–400, 2022.

    • Search Google Scholar
    • Export Citation
  • [15]

    P. Frankl, T. Łuczak, and K. Mieczkowska. On matchings in hypergraphs. Electron. J. Combin., 19(2):article no. 42, 2012.

  • [16]

    E. Friedgut and O. Regev. Kneser graphs are like Swiss cheese. Discrete Anal., page article no. 2, 2018.

  • [17]

    M. Guo and H. Lu. A stability result for almost perfect matchings, 2024. http://arxiv.org/abs/2404.09720v1.

  • [18]

    M. Guo, H. Lu, and Y. Jiang. Improved bound on vertex degree version of Erdős matching conjecture. J. Graph Theory, 104(3):485–498, 2023.

    • Search Google Scholar
    • Export Citation
  • [19]

    J. Han. Perfect matchings in hypergraphs and the Erdős matching conjecture. SIAM J. Discrete Math., 30(3):1351–1357, 2016.

  • [20]

    H. Huang, P.-S. Loh, and B. Sudakov. The size of a hypergraph and its matching number. Combin. Probab. Comput., 21(3):442–450, 2012.

  • [21]

    H. Huang and Y. Zhao. Degree versions of the Erdős-Ko-Rado theorem and Erdős hyper-graph matching conjecture. J. Combin. Theory Ser. A, 150:233–247, 2017.

    • Search Google Scholar
    • Export Citation
  • [22]

    D. J. Kleitman. Maximal number of subsets of a finite set no 𝑘 of which are pairwise disjoint. J. Combinatorial Theory, 5:157–163, 1968.

    • Search Google Scholar
    • Export Citation
  • [23]

    D. Kolupaev and A. Kupavskii. Erdős matching conjecture for almost perfect matchings. Discrete Math., 346(4):article no. 113304, 2023.

  • [24]

    A. Kupavskii. Diversity of uniform intersecting families. European J. Combin., 74:39–47, 2018.

  • [25]

    A. Kupavskii. Degree versions of theorems on intersecting families via stability. J. Combin. Theory Ser. A, 168:272–287, 2019.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896

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Studia Scientiarum Mathematicarum Hungarica
Language English
French
German
Size B5
Year of
Foundation		 1966
Volumes
per Year		 1
Issues
per Year		 4
Founder Magyar Tudományos Akadémia  
Founder's
Address		 H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher Akadémiai Kiadó
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ISSN 0081-6906 (Print)
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