2021/06/30 by Yu‐Lin Chang, Chang, Yulin, Huifen Ge +5 · 1 citation
Mathematics · Computer Science · #Limits and Structures in Graph Theory #Advanced Graph Theory Research #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.2106.16068
For all integers k,d such that k ≥ 3 and k/2≤ d ≤ k-1, let n be a sufficiently large integer \rm(which may not be divisible by k\rm) and let s≤ \lfloor n/k\rfloor-1. We show that if H is a k-uniform hypergraph on n vertices with δd(H)>\binomn-dk-d-\binomn-d-s+1k-d, then H contains a matching of size s. This improves a recent result of Lu, Yu, and Yuan and also answers a question of Kühn, Osthus, and Townsend. In many cases, our result can be strengthened to s≤ \lfloor n/k\rfloor, which then covers the entire possible range of s. On the other hand, there are examples showing that the result does not hold for certain n, k, d and s= \lfloor n/k\rfloor.