Let $\left\{(A_{i},B_{i})\mid i=1,\cdots, m\right\}$ be pairs of sets such that
(1) $|A_{i}|=a$, $|B_{i}|=b$ for all $i=1,\cdots ,m$;
(2) $A_{i}\cap B_{i}=\varnothing$ for all $i=1,\cdots ,m$;
(3) $A_{i}\cap B_{j}\neq \varnothing$ for all $1\leq i,j\leq m$ with $i\neq j$.
Show that $m\leq \left(\begin{array}{c}a+b\\a\end{array}\right)$.
I can find an example that the equality holds. For instance, let $\left\{A_{i}\right\}$ be all the subsets of $\left\{1,\cdots,a+b\right\}$ with exactly $a$ elements and set $B_{i}$ the complement of $A_{i}$. This example also shows that the bounded is the best. However, I have no idea to prove this inequality.
Thanks in advance.