Let $n\geq 3$ be an integer. We call a family ${\cal C}$ of subsets of $\{1,\ldots,n\}$ intersecting if it has the following properties:

1. $A, B\in {\cal C}$ implies $A\cap B \neq \emptyset$, and
2. $A \in {\cal C}, B\subseteq A$ implies $B\notin {\cal C}$.

One way to construct an intersecting family with a large number of elements is the following. For any positive integer $k$ we say ${\cal A}\subseteq {\cal P}(\{1,\ldots,k\})$ is an antichain if no member of ${\cal A}$ is contained in another member of ${\cal A}$. Let ${\cal M}$ be an antichain in ${\cal P}(\{1,\ldots,n-1\})$ with the maximum number of elements amongst all antichains in ${\cal P}(\{1,\ldots,n-1\})$. We denote this number by $m_{n-1}$. Let $${\cal C} = \big\{X\cup\{n\}: X\in {\cal M}\big\}.$$ So ${\cal C}$ is an intersecting family on $ \{1,\ldots,n\}$ with $m_{n-1}$ elements.

Question. Are there intersecting families on on $ \{1,\ldots,n\}$ with more than $m_{n-1}$ elements?

Let $n\geq 3$ be an integer. We call a family ${\cal C}$ of subsets of $\{1,\ldots,n\}$ intersecting if it has the following properties:

1. $A, B\in {\cal C}$ implies $A\cap B \neq \emptyset$, and
2. $A \in {\cal C}, B\subseteq A$ implies $B\notin {\cal C}$.

One way to construct an intersecting family with a large number of elements is the following. For any positive integer $k$ we say ${\cal A}\subseteq {\cal P}(\{1,\ldots,k\})$ is an antichain if no member of ${\cal A}$ is contained in another member of ${\cal A}$. Let ${\cal M}$ be an antichain in ${\cal P}(\{1,\ldots,n-1\})$ with the maximum number of elements amongst all antichains in ${\cal P}(\{1,\ldots,n-1\})$. We denote this number by $m_{n-1}$. Let $${\cal C} = \big\{X\cup\{n\}: X\in {\cal M}\big\}.$$ So ${\cal C}$ is an intersecting family on $ \{1,\ldots,n\}$ with $m_{n-1}$ elements.

Question. Are there intersecting families on $ \{1,\ldots,n\}$ with more than $m_{n-1}$ elements?