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 4 of 5: Enlarge the set
Detailed analysis
Since u belongs to no group with two members in S, adding u cannot create a group properly contained in S′: any group contained in S′ and using u would need two such members. Groups already contained in S′ were absent from S. Therefore S′ has the required property and is larger than S, contradicting maximality.