Ramsey Theory
Frank Plumpton Ramsey was a British mathematician, who lived in the early 20th century and made major contributions before his premature death at the age of 26.
Ramsey theory is a branch of mathematics that deals with structures, particularly graphs, seeking to understand the emergence of order within seemingly random systems.
Here we illustrate a result from Ramsey Theory. How large should a group of people be in order to guarantee that either there are three individuals who know each other or three who are strangers to each other?
The vertices in the graph represent people, while edge colours indicate friendship status. So we are looking for triangles of the same colour.
We have a pentagon and an hexagon. The vertices represent people, and the colour of the edge connecting them indicates whether they are friends or strangers. You can change the edge colour by gently rotating it.
Attempt adjusting the edges to avoid forming any monochromatic
triangles! This is feasible on just one of the boards.
Aim for one blue triangle and no red ones. Achievable on only
one board as well.
On the hexagon, if you are obliged to have at least 7 of the
edges blue and 8 red, what is the largest number of
monochromatic triangles you can find?
The answer is 9. Can you create such a figure?