Question 116
In a university entrance examination with $2006000$ student participants, each student makes a list of $12$ colleges from among a total of $2006$ colleges. It turns out that, for any $6$ students, there exist two colleges such that each of these $6$ students has included at least one of these colleges into his (her) list. A list, which includes at least one college from each student's list, is called an *extensive list*.
a. Prove that there exists an extensive list with $12$ elements whatever the students' lists are, so long as they satisfy the given conditions.
b. Show that there exist students' lists satisfying the given conditions such that no list shorter than $12$ elements is extensive.
a. Prove that there exists an extensive list with $12$ elements whatever the students' lists are, so long as they satisfy the given conditions.
b. Show that there exist students' lists satisfying the given conditions such that no list shorter than $12$ elements is extensive.