Stability on Matchings in 3-Uniform Hypergraphs.

Given a positive integer r, let [ r ] = { 1 , … , r } . Let n, m be positive integers such that n is sufficiently large and 1 ≤ m ≤ ⌊ n / 3 ⌋ - 1 . Let H be a 3-graph with vertex set [n], and let δ 1 (H) denote the minimum vertex degree of H. The size of a maximum matching of H is denoted by ν (H) . Kühn, Osthus and Treglown (2013) proved that there exists an integer n 0 ∈ N such that if H is a 3-graph with n ≥ n 0 vertices and δ 1 (H) > n - 1 2 - n - m 2 , then ν (H) ≥ m . In this paper, we show that there exists an integer n 1 ∈ N such that if | V (H) | ≥ n 1 , δ 1 (H) > n - 1 2 - n - m 2 + 3 and ν (H) ≤ m , then H is a subgraph of H ∗ (n , m) , where H ∗ (n , m) is a 3-graph with vertex set [n] and edge set E (H ∗ (n , m)) = { e ⊆ [ n ] : | e | = 3 and e ∩ [ m ] ≠ ∅ } . The minimum degree condition is best possible. [ABSTRACT FROM AUTHOR]