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Normal tic-tac-toe can always be drawn. But what if it lives in high dimensions? It turns out that no matter how large a tic-tac-toe board you have or how many players want to play, there always is a dimension long enough that guarantees the 1st player will always win. The theorem behind this, Hales-Jewett, is part of a family of theorems in Ramsey theory that show how lower level structures (like straight lines of the same colour) are always going to occur if the dimension is large enough. That is, you can’t have a system that is totally without order. In this video we explore these tic-tac-toe generalizations, the Van Der Waerden theorem and sketch it’s proof.
Reference: This undergrad level book introducing Ramsey Theory has lots more detail on all the theorems and more: ~vjungic/RamseyNotes/
0:00 Friends and Strangers Theorem
2:56 What is Ramsey Theory?
3:36 High dimensional Tic-Tac-Toe
7:58 Hales-Jewett Theorem
10:37 Van der Waerden’s theorem
14:42 Proof sketch of Van der Waerden’s theorem
21:23 Summary
21:57
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