g If every pair of people in a group are friends, or if every pair are strangers, the group is called uniform. Show the above results imply that every set of 6 people includes two uniform three-person groups.

Question

Maverick Raymond

Mathematics

11 July, 09:48

g If every pair of people in a group are friends, or if every pair are strangers, the group is called uniform. Show the above results imply that every set of 6 people includes two uniform three-person groups.

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Answers (1)

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Leah Travis · Answer 1

Every group of 6 people has at least two uniform 3-person groups.

Step-by-step explanation:

Denote the 6-people by 6-vertices and draw a blue edge between 2-edges. If the two persons representing the vertices are friends. Otherwise, draw a red edge. This gives rise to a colored graph K6, edges arecolored with either blue or red.

There can exist at most 36 mips. so, multicolor triangles can exist at most 36/2 = 18 multicolor triangles.

If there are 20-triangle in graph. Therefore, every graph of 6-people has at least two uniform 3-person groups.