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

For a positive integer $d\ge 2$, a family $\mathcal{F}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ is said to be d-wise intersecting if $\mathrm{\mid }{F}_1\cap F_2\cap \cdots \cap F_d\mathrm{\mid }\ge 1$ for all $F_1,F_2,\dots ,F_d\in \mathcal{F}$. A d-wise intersecting family $\mathcal{F}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ is called maximal if $\mathcal{F}\cup \{A\}$ is not d-wise intersecting for any $A\in \left(\genfrac{}{}{0px}{}{[n]}{k}\right)\setminus \mathcal{F}$. We provide a refinement of O’Neill and Verstraëte’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 $k>d+1\ge 4$, and the fifth largest and the sixth largest maximal non-trivial 3-wise intersecting k-uniform families for any $k\ge 5$, in the asymptotic sense. Our proofs are applications of the $\mathrm{\Delta }$-system method.