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Keep exploring at ► Get started for free for 30 days — and the first 200 people get 20% off an annual premium subscription! 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 Check out my MATH MERCH line in collaboration with Beautiful Equations ► COURSE PLAYLISTS: ►DISCRETE MATH: ►LINEAR ALGEBRA: ►CALCULUS I: ► CALCULUS II: ►MULTIVARIABLE CALCULUS (Calc III): ►VECTOR CALCULUS (Calc IV) ►DIFFERENTIAL EQUATIONS: ►LAPLACE TRANSFORM: ►GAME THEORY: OTHER PLAYLISTS: ► Learning Math Series ►Cool Math Series: BECOME A MEMBER: ►Join: MATH BOOKS I LOVE (affilliate link): ► SOCIALS: ►Twitter (math based): ►Instagram (photography based):
