Fractional and integer matchings in uniform hypergraphs

Abstract

Abstract. A conjecture of Erdős from 1965 suggests the minimum number of edges in a kuniform hypergraph on n vertices which forces a matching of size t, where t ≤ n/k. Our main result verifies this conjecture asymptotically, for all t < 0.48n/k. This gives an approximate answer to a question of Huang, Loh and Sudakov, who proved the conjecture for t ≤ n/3k2 . As a consequence of our result, we extend bounds of Bollobás, Daykin and Erdős by asymptotically determining the minimum vertex degree which forces a matching of size t < 0.48n/(k − 1) in a k-uniform hypergraph on n vertices. We also obtain further results on d-degrees which force large matchings. In addition we improve bounds of Markström and Ruciński on the minimum ddegree which forces a perfect matching in a k-uniform hypergraph on n vertices. Our approach is to inductively prove fractional versions of the above results and then translate these into integer versions.

Near Perfect Matchings in ${k}$-Uniform Hypergraphs II

“…By the previous results [6,16], it seems that m ′ d (k, n) is determined only by the space barrier. However, we give the following divisibility barrier construction which generalizes Construction 1.3, showing that this is not always the case when d ≤ k − 2.…”

“…Bollobás, Daykin and Erdős [2] determined m s 1 (k, n) for s < n/2k 3 − 1. Recently, Kühn, Osthus and Townsend [16] determined m s d (k, n) asymptotically for 1 ≤ d ≤ k − 2 and s ≤ min{n/2(k − d), (1 − o(1))n/k}. 1.2.…”

Perfect Matchings in Hypergraphs and the Erdös Matching Conjecture

“…The equality was shown for d = k − 1 in [25] (in fact, it is shown that f n/k k−1 (k, n) = ⌈n/k⌉), and for k/2 ≤ d < k − 1 in [18].…”

“…Here we use the approach in [18] as well as two propositions. In fact, we only replace their [18, Theorem 1.8] by our Theorem 1.8.…”

A Note on Perfect Matchings in Uniform Hypergraphs

“…Alon, Frankl, Huang, Rödl, Ruciński and Sudakov [1] determined m ℓ (k, n) asymptotically in the case when (k, ℓ) = (5, 1), (5,2), (6,2), and (7,3). Other than these results, no other asymptotic or exact results are known (the best known general bounds are due to Kühn, Osthus and Townsend [11]).…”