Friends and Strangers

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Deals with not being able to trust people, who claim to be your friends.

Friends and strangers

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The Problem. The MC. The Soup. By continuing you agree to the use of cookies. Get Access Get Access. Author links open overlay panel Markus Freitag a Paul. Bauer b. Abstract Research on the foundations of social trust mainly concentrates on the evaluation of one's social environment. Keywords Big Five.

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What about R a,b for general values of a and b? Can we be sure it even exists? Perhaps no number of people is enough to guarantee a friends and b strangers. But luckily we can be sure — this result is known as Ramsey's theorem.

Friends and Strangers

It tells us that however much orderliness we want, we can find it as long as the graph we are given is big enough. So what is the value of R 5,5? The answer is: nobody knows! Very few Ramsey numbers R a,b are actually known with a and b both bigger than 2. The most we can say about R 5,5 with our present knowledge is that it is somewhere between 42 and How can it be so difficult to get an accurate value? Arguments involve finding upper bounds — for example, we saw quite easily that R 3,3 could not be bigger than 6. But to show that it could not be as small as 5, we had to construct a graph with 5 points, as a counterexample.

Friends and Strangers

The problem is that we are looking for examples of order, so the best counterexamples will usually have lots of disorder — they will look random. This makes it hard or impossible to find a "rule" that gives good counterexamples.

Anything constructed by rule will probably have too much order in it. Also, our upper bounds may be too high, but how will we ever prove it? Perhaps by examining all possible graphs on a computer to show that one has the right number of friends or strangers?

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The problem with this approach is that numbers explode rapidly. To show that R 5,5 is at most 49 we would have to look at 2 possible colourings of a graph. This number is far, far bigger than the number of particles in the known Universe.