Determine the maximal size of a set of positive integers with the following properties:

(1) The integers consist of digits from the set $\{1,2,3,4,5,6\}$.

(2) No digit occurs more than once in the same integer.

(3) The digits in each integer are in increasing order.

(4) Any two integers have at least one digit in common (possibly at different positions).

(5) There is no digit which appears in all the integers.

Answer: 32.
Associate with any $a_{i}$ the set $M_{i}$ of its digits. By (??), (??) and (??) the numbers are uniquely determined by their associated subsets of $\{1,2, \ldots, 6\}$. By (??) the sets are intersecting. Partition the 64 subsets of $\{1,2, \ldots, 6\}$ into 32 pairs of complementary sets $(X,\{1,2, \ldots, 6\}-X)$. Obviously, at most one of the two sets in such a pair can be a $M_{i}$, since the two sets are non-intersecting. Hence, $n \leq 32$. Consider the 22 subsets with at least four elements and the 10 subsets with three elements containing 1 . Hence, $n=32$. 7. A photographer took some pictures at a party with 10 people. Each of the 45 possible pairs of people appears together on exactly one photo, and each photo depicts two or three people. What is the smallest possible number of photos taken?

Answer: 19.