Problem 2
Students in a class form groups, each of which contains exactly three members, such that any two distinct groups have at most one member in common. Prove that, when the class size is 46, there is a set of 10 students in which no group is properly contained.
Step 1 of 5: Choose a maximal set
Detailed analysis
Let C be the 46 students and choose a set S of maximum size that contains no group properly. It is enough to prove that s=|S| is at least 10, because any 10-element subset of S then has the same property.