A Note on Exact Minimum Degree Threshold for Fractional Perfect Matchings

Rodl, Rucinski, and Szemeredi determined the minimum (k - 1)-degree threshold for the existence of fractional perfect matchings in k-uniform hypergrahs, and Kiihn, Osthus, and Townsend extended this result by asymptotically determining the d-degree threshold for the range k - 1 > d >= k/2. In this note, we prove the following exact degree threshold: let k, d be positive integers with k >= 4 and k - 1 > d >= k/2, and let n be any integer with n >= 2k (k - 1) + 1. Then any n-vertex k-uniform hypergraph with minimum d-degree delta(d) (H) > [GRAPHICS] - [GRAPHICS] contains a fractional perfect matching This lower bound on the minimum d-degree is best possible. We also determine the minimum d-degree threshold for the existence of fractional matchings of size s, where 0 < s <= n/k (when k/2 <= d <= k - 1), or with s large enough and s <= n/k (when 2k/5 < d < k/2).