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 <= left perpendicular n/3 right perpendicular - 1. Let H be a 3-graph with vertex set [n], and let delta(1) (H) denote the minimum vertex degree of H. The size of a maximum matching of H is denoted by v(H). Kuhn, Osthus and Treglown (2013) proved that there exists an integer n(0) is an element of N such that if H is a 3-graph with n >= n(0) vertices and delta(1) (H) > (n-m(2)) - (n-m(2)), then v(H) >= m. In this paper, we show that there exists an integer n(1) is an element of N such that if vertical bar V(H)vertical bar >= n(1), delta(1) (H) > (n-1(2)) - (n-m(2)) + 3 and v(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 subset of [n] : vertical bar e vertical bar = 3 and e boolean AND [m] not equal empty set}. The minimum degree condition is best possible.