Claims

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1961_01_01_erdos_ko_rado: Erdős, Ko and Rado (1961) prove that an intersecting family of r-subsets of an n-set with n at least 2r has at most binomial(n-1, r-1) members, which is the matching conjecture for k = 2 and every uniformity; refereed.

1965_01_01_erdos: Erdős (1965) proves the conjectured value of f(n; r, k) whenever n is at least c_r k, for a constant c_r depending only on r, the range in which the covering term dominates; refereed, with the constant unspecified.

1968_09_01_kleitman: Kleitman (1968) bounds families of subsets of an n-set with no k pairwise disjoint members, sharply when k divides n or n + 1; the r-uniform case at n = rk is the matching conjecture there; refereed.

1976_01_01_bollobas_daykin_erdos: Bollobás, Daykin and Erdős (1976) prove that for n > 2 r^3 (k - 1) an r-uniform hypergraph with no k pairwise disjoint edges and more edges than the covering family minus a correction is contained in it; refereed.

2011_07_27_huang_loh_sudakov: Huang, Loh and Sudakov (2012) prove the matching conjecture whenever n > 3 r^2 k, by shifting and through an asymptotic rainbow-matching strengthening; refereed in Combin. Probab. Comput.

2012_02_02_frankl_rodl_rucinski: Frankl, Rödl and Ruciński (2012) prove the matching conjecture for 3-uniform hypergraphs whenever n is at least four times the forbidden matching size, that is n at least 4k; refereed in Combin. Probab. Comput.

2012_02_19_luczak_mieczkowska: Łuczak and Mieczkowska (2014) prove the matching conjecture for 3-uniform hypergraphs on n vertices for every matching number once n is large, with the cover and the clique the only extremal hypergraphs; refereed.

2012_05_30_frankl: Frankl (2017) proves the matching conjecture for 3-uniform hypergraphs for every n and every matching number, completing the case r = 3; refereed in Discrete Applied Mathematics.

2012_06_13_frankl_luczak_mieczkowska: Frankl, Łuczak and Mieczkowska (2012) prove that an r-uniform hypergraph whose largest matching has k - 1 edges has at most the covering family's size once n > 2 r^2 (k - 1) / log r, the cover alone extremal; refereed.

2013_07_01_frankl: Frankl (2013) proves the matching conjecture whenever n is at least (2s + 1) r - s for the matching number s = k - 1, the first range linear in both parameters; refereed in J. Combin. Theory Ser. A.

2017_10_01_frankl: Frankl (2017) proves the matching conjecture when k - 1 > r and n lies in rk <= n < k(r + 1/(2 r^(2r+1))), the almost-perfect range where the clique term is the maximum; refereed in Israel J. Math.

2018_06_22_frankl_kupavskii: Frankl and Kupavskii (2022) prove the matching conjecture for large matching number s = k - 1 whenever n is at least (5/3) s r - (2/3) s, by concentration inequalities; refereed in J. Combin. Theory Ser. B.

2022_06_03_kolupaev_kupavskii: Kolupaev and Kupavskii (2023) prove the matching conjecture for r >= 5 and k - 1 > 101 r^3 whenever rk <= n < k(r + 1/(100 r)), widening Frankl's window above n = rk; refereed in Discrete Mathematics.

2026_02_01_mishra: Mishra's 2026 preprint claimed a complete proof of the Erdős matching conjecture; a reader located an error in its Lemma 4 on the site's thread, and the author withdrew the preprint on 1 June 2026.

2026_02_22_frankl_lu_ma_wu: Frankl, Lu, Ma and Wu (2026) claim the matching conjecture for 4-uniform hypergraphs whenever n is at least five times the matching number k - 1 and n is large, through a stability theorem; an arXiv preprint, not refereed.

2026_05_25_hou_hu_liu: Hou, Hu and Liu (2026) claim the matching conjecture for 4-uniform hypergraphs for every matching number s at least 6004 and n at least 4s + 4, by a finite-board reduction checked in exact arithmetic; not refereed.

2026_08_19_cao_liu_zhang: Cao, Liu and Zhang (2026) claim the matching conjecture for every fixed uniformity r once the matching number k - 1 is large and n is at least (r + 1)(k - 1), the conjecture's crossover; an arXiv preprint, not refereed.

2026_09_13_babanskyy: Babanskyy's 2026 manuscript claiming the Erdős matching conjecture for 4-uniform hypergraphs and every matching size, by the finite-board method with rational certificates; AI-assisted, unreviewed, not refereed.

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