TW-TST 1997 Problem 9

For $n\geq k\geq 3$ , let $X=\{1,2,...,n\}$ and let $F_{k}$ a the family of $k$ -element subsets of $X$ , any two of which have at most $k-2$ elements in common. Show that there exists a subset $M_{k}$ of $X$ with at least $[\log_{2}{n}]+1$ elements containing no subset in $F_{k}$ .