讲座内容：

The covering number of a family $\mathcal{F}$ is the size of the smallest set that intersects every set in the family, denoted by $\tau(\mathcal{F})$.

For $1 \leq p \leq k-2$, define

\[

\mathcal{B}_p = \{[2,k] \cup \{j\} : k+1 \leq j \leq k+p+1\} \cup \{[k+1, 2k]\},

\]

\[

\mathcal{A}_p = \left\{ A \in \binom{[n]}{k} : 1 \in A, \text{ and } A \cap B \neq \emptyset \text{ for each } B \in \mathcal{B}_p \right\}

\]

and

\[

\mathcal{G}_p(n,k) = \mathcal{A}_p \cup \mathcal{B}_p.

\]

In this talk, we prove that for $n > 2k$ and $k \geq 4$, if $\mathcal{F}$ is a maximal intersecting family in $\binom{[n]}{k}$ with $\tau(\mathcal{F}) \geq 3$ and

\[

|\mathcal{F}| \geq |\mathcal{G}_2(n,k)|,

\]

then $\mathcal{F} \cong \mathcal{G}_1(n,k)$ or $\mathcal{F} \cong \mathcal{G}_2(n,k)$.

This generalizes the results obtained by Kupavskii, Frankl and Wang, respectively.