Let $n>1$ be an integer and let $[n]=\{1,\ldots,n\}$. An *intersecting family* on $[n]$ is a set $E\subseteq {\cal P}([n])$ such that for all $a,b\in E$ we have $a\cap b\neq\emptyset$. It is easy to see that an intersecting family on $[n]$ can have size $2^{n-1}$: fix $j\in[n]$ and let $E = \{s\subseteq [n]:j\in s\}$. 


**Question 1.** If $E$ is an intersecting family on $[n]$ with $|E| = 2^{n-1}$, is there necessarily $j\in [n]$ such that $E = \{s\subseteq [n]:j\in s\}$?


**Question 2.** If $T\subseteq {\cal P}([n])$ has the property that $|T| > 2^{n-1}$, does this imply that $T$ is not intersecting?



Size of intersecting families on $\{1,\ldots,n\}$
For $n > 2$ the answer to question 1 is no. If $n$ is odd take $E$ to be all subsets of size greater than $n/2$, and if $n$ is even take all subsets of size greater than $n/2$ along with all sets of size $n/2$ that *exclude* a certain element.


