Mathematician Frank Ramsey showed how to discover coherent patterns among a multitude of number groupings
Imagine you are hosting a small party for six guests, and it isn’t understood who knows one another and who doesn’t. As it turns out, there’s bound to be at least three people who are complete strangers to one another—or three who are already friends. So there will always be at least one group of three people who are all either known or completely unknown to one another.
The whole thing can be solved with figures, or graphs, which are networks of points and edges . Each person symbolizes a node. The six guests can be arranged in a circle. Now each point is connected. This creates the 15 edges. Depending on whether two people know each other or not, they are colored red or blue . Now the claim: no matter how you color the edges, you always get a monochromatic triangle—an all-blue or all-red figure.
With four people, it is also easy to find cases in which there is no group of three who re all unknown to or else friends with one another. The hostess is friends with at least three people—or at least three people are unknown to her. In a graph, this can be seen by the fact that she always has at least three edges of the same color. Using a concrete example, one can assume that, in one of the configurations shown in the table, the hostess has three red edges, meaning she knows three other guests.
It has been unclear for decades how large R is, however. In other words, what is the smallest number of guests that you have to invite so that there is always a five-person grouping of all acquaintances or strangers? Experts have been able to narrow down the result: we now know that R falls within a range that is less than or equal to 43 nodes on the lower end and less than or equal to 48 nodes at the upper boundary.
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