KMO 2015 Problem 7

A positive integer $n$ is given. If there exists sets $F_1, F_2, \cdots F_m$ satisfying the following conditions, prove that $m \le n$ . (For sets $A, B$ , $|A|$ is the number of elements of $A$ . $A-B$ is the set of elements that are in $A$ but not $B$ . $\text{min}(x,y)$ is the number that is not larger than the other.)

(i): For all $1 \le i \le m$ , $F_i \subseteq \{1,2,\cdots,n\}$ (ii): For all $1 \le i < j \le m$ , $\text{min}(|F_i-F_j|,|F_j-F_i|) = 1$