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.

Improved bound on vertex degree version of Erdős matching conjecture

“…For a 3‐graph $$H$$, Hàn, Person, and Schacht [9] showed that $${\delta }_{1}(H)\gt (5\unicode{x02215}9+o(1))\left(\genfrac{}{}{0ex}{}{|V(H)|}{2}\right)$$ is sufficient for the appearance of a perfect matching of $$H$$. Kühn, Osthus, and Townsend [16] proved a stronger result: There exists a positive integer $${n}_{0}$$ such that if $$H$$ is a 3‐graph with $$|V(H)|=n\ge {n}_{0}$$, $$m$$ is an integer with $$1\le m\le n\unicode{x02215}3$$, and $${\delta }_{1}(H)\gt \left(\genfrac{}{}{0ex}{}{n-1}{2}\right)-\left(\genfrac{}{}{0ex}{}{n-m}{2}\right)$$, then $$\nu (H)\ge m$$. For $$k\in \{3,4\}$$, Khan [12, 13] showed that there exists a positive integer $${n}_{0}$$ such that if $$H$$ is a …”

“…is sufficient for the appearance of a perfect matching of H . Kühn, Osthus, and Townsend [16] proved a stronger result: There exists a positive integer n 0 such that if H is a 3-graph with…”