The Erdős Matching Conjecture states that the maximum size 𝑓 (𝑛, 𝑘, 𝑠) of a family
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
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.
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.
S. Das and T. Tran. Removal and stability for Erdős-Ko-Rado. SIAM J. Discrete Math., 30(2):1102–1114, 2016.
P. Erdős. A problem on independent 𝑟-tuples. Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 8:93–95, 1965.
P. Erdős, Chao Ko, and R. Rado. Intersection theorems for systems of finite sets. Quart. J. Math. Oxford Ser. (2), 12:313–320, 1961.
P. Frankl. Erdős-Ko-Rado theorem with conditions on the maximal degree. J. Combin. Theory Ser. A, 46(2):252–263, 1987.
P. Frankl. Extremal set systems. In Handbook of combinatorics, Vol. 1, 2, pages 1293–1329. Elsevier Sci. B. V., Amsterdam, 1995.
P. Frankl. Improved bounds for Erdős’ matching conjecture. J. Combin. Theory Ser. A, 120(5):1068–1072, 2013.
P. Frankl. On the maximum number of edges in a hypergraph with given matching number. Discrete Appl. Math., 216(3):562–581, 2017.
P. Frankl. Proof of the Erdős matching conjecture in a new range. Israel J. Math., 222(1):421–430, 2017.
P. Frankl. Maximum degree and diversity in intersecting hypergraphs. J. Combin. Theory Ser. B, 144:81–94, 2020.
P. Frankl. On non-trivial families without a perfect matching. European J. Combin., 84:article no. 103044, 2020.
P. Frankl and A. Kupavskii. Diversity. J. Combin. Theory Ser. A, 182:article no. 105468, 2021.
P. Frankl and A. Kupavskii. The Erdős matching conjecture and concentration inequalities. J. Combin. Theory Ser. B, 157:366–400, 2022.
P. Frankl, T. Łuczak, and K. Mieczkowska. On matchings in hypergraphs. Electron. J. Combin., 19(2):article no. 42, 2012.
E. Friedgut and O. Regev. Kneser graphs are like Swiss cheese. Discrete Anal., page article no. 2, 2018.
M. Guo and H. Lu. A stability result for almost perfect matchings, 2024. http://arxiv.org/abs/2404.09720v1.
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.
J. Han. Perfect matchings in hypergraphs and the Erdős matching conjecture. SIAM J. Discrete Math., 30(3):1351–1357, 2016.
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.
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.
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.
D. Kolupaev and A. Kupavskii. Erdős matching conjecture for almost perfect matchings. Discrete Math., 346(4):article no. 113304, 2023.
A. Kupavskii. Diversity of uniform intersecting families. European J. Combin., 74:39–47, 2018.
A. Kupavskii. Degree versions of theorems on intersecting families via stability. J. Combin. Theory Ser. A, 168:272–287, 2019.