$S$ be a collection of subsets of $\{1,...,100\}$ ; any two sets in $S$ has non-empty intersection , what is the maximum possible value of $|S|$?

Solution 1:

Consider the collection $S_1$ of all subsets that contain the number $1.$ It satisfies the condition and its cardinality is $2^{99}.$

On the other hand let $S$ be such a collection and consider the partition into two subcollections $S_y$ and $S_n$ of sets according to whether they do, or do not, contain the number $1.$

$S_n$ has at most $2^{99}$ elements because those elements are subsets of $\{2,\ldots,100\}.$

But $S_y$ cannot contain the complement of any set in $S_n$, which rules out exactly $2^{99}-|S_n|$ possibilities.

Therefore $S=S_y\cup S_n$ has at most $2^{99}$ elements.