组合数学 (Spring 2013)/Problem Set 2

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  • 每道题目的解答都要有完整的解题过程。中英文不限。
  • 这次作业只有一个星期的时间。

Problem 1

Prove the following identity:

  • .

(Hint: Use double counting.)

Problem 2

Show that among any group of people, where , there are at least two people who know exactly the same number of people in the group (assuming that "knowing" is a symmetric relation).

Problem 3

Let be a subset of such that . Show that there exist such that and are coprime.

Problem 4

(Due to Karger)

Balls of 8 different colors are in 6 bins. There are 20 balls of each color. Prove that there must be a bin containing 2 pairs of balls from the two different colors of balls (like {red,red,blue,blue}).

Problem 5

(Erdős-Spencer 1974)

Let coins of weights 0 and 1 be given. We are also given a scale with which we may weigh any subset of the coins. Our goal is to determine the weights of coins (that is, to known which coins are 0 and which are 1) with the minimal number of weighings.

This problem can be formalized as follows: A collection is called determining if an arbitrary subset can be uniquely determined by the cardinalities .

  • Prove that if there is a determining collection , then there is a way to determine the weights of coins with weighings.
  • Use pigeonhole principle to show that if a collection is determining, then it must hold that .

(This gives a lower bound for the number of weighings required to determine the weights of coins.)