Related Numbers of a Set

★ 9/10combinatorics Counting Set Theory Pigeonhole

Problem

Let

A_{2n} = \{1, 2, 3, \ldots, 2n\} \qquad (n \in \mathbb{N}^*,\ n \geqslant 2).

Call a positive integer $m$ ( $m \geqslant 4$ ) a related number of $A_{2n}$ if every $m$ -element subset $P$ of $A_{2n}$ contains $4$ elements whose sum is $4n + 1$ .

1. For $n = 3$ : decide whether $5$ and $6$ are related numbers of $A_6$ , with reasons. 2. If $m$ is a related number of $A_{2n}$ , prove that $m - n - 3 \geqslant 0$ . 3. For a given $n$ , find the minimum related number $m$ of $A_{2n}$ .

Answer

Solution

Difficulty	9/10
Topics	combinatorics, Counting, Set Theory, Pigeonhole

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