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Fun with Ramsey Theory!
(Taught in Splash 2015)
In the 1950s, a Hungarian sociologist noticed that in every set of around twenty children, there were four who were all friends, or four none of whom were friends. Had he just discovered a remarkable property of human interactions? No! Enter Ramsey theory which explores results about some partition of a set maintaining the property of the whole. How big of a complete graph do we need for any partition of its edges into two "colors" to contain a monochromatic k-clique? What's the smallest n such that any way of splitting the numbers 1,2,...,n into two disjoint sets guarantees that one of your two sets contains an arithmetic progression of length 4? Explore the answers to these and other fun questions! I'll even provide some open problems for you to think about afterward.