Probability and set theory

Question

Consider sets $A_1,A_2,A_3 \text{ to } A_m \text{ where }A_i \subseteq [n] \text{ and } [n] = \{1,2,3, \dots n \}$ such that there are no three distinct sets in the collection with the property $A_i \subset A_j \subset A_k.$ For even $n$, prove that

\begin{align*}
m \leq 2\cdot \binom{n}{\frac{n}{2}}
\end{align*}

 

For even $n$ prove a more stronger bound on $m$ :

\begin{align*}
m \leq \binom{n}{\frac{n}{2}} + \binom{n}{\frac{n}{2} - 1}
\end{align*}