29.2.2. Ramsay Numbers

Try these first

1. 1.

   What is a Ramsay number?

   Hint

   Think about groups and relationships, specifically friendships or rivalries.

2. 2.

   What does it mean for two nodes in a graph to be connected?

   Hint

   Remember, an edge signifies friendship in our model.

3. 3.

   What does the Ramsay number R(3, 3) signify?

   • It guarantees 3 mutual friends
   • It guarantees 3 mutual enemies
   • It guarantees both

   Hint

   Think about how friendship and rivalry can be interchanged.

4. 4.

   True or False: Every group of 6 people will always contain either three mutual friends or three mutual enemies.

   • True
   • False

   Hint

   This is the foundational claim of Ramsay's theory.

5. 5.

   In a group of 6 individuals A, B, C, D, E, F, construct a diagram to show all possible pair relationships (friend or enemy), and demonstrate Rogers' theorem seat structure.

   Hint

   Remember your basic graphing skills as you depict relationships.

6. 6.

   Prove the Ramsey theorem by finding subsets of the group above and showing there will always be either three friends or three enemies.

   Hint

   Use both the original and complement graphs.