Sunday, March 27, 2016

Various double counting problems

These problems can all be solved by the double counting principle, that is, to count the same object two different ways and assert that those must match.

Suppose $n > 4$. Let $A_1, A_2, \dots ,A_k$ each be a set of $n$ integers, where $k \leq 4^{n-1} / 3^n$. Show that we can color each integer with one of the four colors so that each set contains integers of 4 different colors.

In a party attended by $n$ people, there were a number of handshakes. A group of 10 persons is called "tight" if each person shook hands with the other 9. It is known that we cannot add another handshake (two people who didn't shake hands now shaking hands) without increasing the number of tight groups. Prove that the number of handshakes is at least $8n-36$

In a class with many students, we choose 2016 committees among them. Each committee has 11 students, and each student may belong to multiple committees. Prove that there is a way to choose some of the students so that each committee has at least one chosen and one unchosen student.

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