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 3 of 5: Find an uncovered student
Detailed analysis
There are 46−s≥37 students outside S, more than the at most 36 students covered by such groups. Thus some student u outside S belongs to no group containing two students of S.