The structure of maximal non-trivial d-wise intersecting uniform families with large sizes

For a positive integer $d\geq 2$, a family $\mathcal F\subseteq \binom{[n]}{k}$ is said to be d-wise intersecting if $|F_1\cap F_2\cap \dots\cap F_d|\geq 1$ for all $F_1, F_2, \dots ,F_d\in \mathcal F$. A d-wise intersecting family $\mathcal F\subseteq \binom{[n]}{k}$ is called maximal if $\mathcal F\cup\{A\}$ is not d-wise intersecting for any $A\in\binom{[n]}{k}\setminus\mathcal F$. We provide a refinement of O'Neill and Verstra\"{e}te's Theorem about the structure of the largest and the second largest maximal non-trivial d-wise intersecting k-uniform families. We also determine the structure of the third largest and the fourth largest maximal non-trivial d-wise intersecting k-uniform families for any $k>d+1\geq 4$, and the fifth largest and the sixth largest maximal non-trivial 3-wise intersecting k-uniform families for any $k\geq 5$, in the asymptotic sense. Our proofs are applications of the $\Delta$-system method.