Triple systems with bounded matching number: some constructions and exact Tur\'{a}n number

We study the Tur\'{a}n numbers of $3$-graphs avoiding $3$-graphs $F$ and $M_{s+1}^3$, a matching of size $s+1$. We disprove a conjecture of Gerbner, Tompkins, and Zhou [European Journal of Combinatorics, 2025, 127:104155] on $\ex(n,\{F,M^3_{s+1}\})$ for $3$-graph $F$ with $\chi(F)=2$ by constructing infinitely many counterexamples. For this family, we determine the asymptotic Tur\'{a}n number via edge-colored Tur\'{a}n problem. In addition, for the $3$-graph $F_{3,2}$ with edge set $\{123,145,245,345\}$, we determine the exact value of $\ex(n,\{F_{3,2}, M_{s+1}^3\})$ for every integers $s$ and all $n \ge 12s^2$.