# (The party problem). Prove the following: At a party of six people there must exist either three people who have all met one another or three people who are mutual strangers.

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(The party problem). Prove the following: At a party of six people there must exist either three people who have all met one another or three people who are mutual strangers.

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Step 1

The above problem is known as Friendship Theorem and is an example of a Ramsey theory.                                                                                                                                                                         Imagine each person is a vertex of a graph.

Draw a blue line between friends and a red line between people who are not strangers.                                                                                                                                                                                                   From one vertex, we can have 0,1,2,3,4,5 blue lines, which is accompanied by 5,4,3,2,1,0 red lines.

Step 2

We either have 3+ blue lines or 3+ red lines.

Case 1: - For 3+ blue lines look for the friends

If any are friends, these are 3 mutual friends (blue triangle) as below: -

Step 3

If none are friends, these are 3 mutual st...

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