Triple Systems With Bounded Matching Number: Some Constructions and Exact Turán Number

ABSTRACT We study the Turán numbers of 3‐graphs avoiding 3‐graphs and , a matching of size . We disprove a conjecture of Gerbner, Tompkins, and Zhou [ European Journal of Combinatorics , 2025, 127:104155] on for a 3‐graph with by constructing infinitely many counterexamples. For this family, we determine the asymptotic Turán number via an edge‐colored Turán problem. We also construct an infinite positive family of 3‐graphs for the conjecture and determine the asymptotic extremal value. In addition, for the important 3‐graph with edge set , we determine the exact value of for every integer and all . The main results are precise formulas for it, accompanied by a relatively concise proof.