Hypergraphs and Acute Triangles | A Surprising Connection! | #SoMEpi

We strengthen a problem about acute triangles from the 1970 International Math Olympiad (Problem 6). Namely, we improve the bound from 70% to 56.15%. The solution uses recent results in the theory of extremal hypergraphs. We connect the problem on acute triangles with a hypergraph conjecture called Turán's tetrahedron conjecture. This video is my submission to the Summer of Math Exposition, pi edition (#SoMEpi): https://some.3b1b.co My video won an Honorable Mention! To stay up to date, subscribe to this YouTube channel. Thanks! 00:00 Introduction 01:23 (Extremal) Graph Theory 04:52 Mantel's Theorem on Triangle-Free Graphs 07:48 Hypergraphs 12:40 Turán's Tetrahedron Conjecture 19:08 Acute Triangles and Hypergraphs 23:00 Construction with Many Acute Triangles 24:12 Open Problem My previous video on the acute triangle problem (with a 2/3 upper bound): https://youtu.be/QCWxegp9MXc Proof of Mantel's theorem (via counting subgraphs) by J. A. Bondy (1997): https://www.sciencedirect.com/science/article/pii/S0012365X96001628 Probabilistic proof of Turán's theorem in graph theory: https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_theorem#Probabilistic_Method The Wikipedia page says the proof is due to Noga Alon and Joel Spencer, but actually I (Ravi Boppana) discovered the proof around 1987 and showed it to Joel. The Erdős–Stone–Simonovits theorem: https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Stone_theorem The Turán Numbers up to 13: Thomas H. Spencer (1993), "On the Size of Independent Sets in Hypergraphs", in "Coding Theory, Design Theory, Group Theory: Proceedings of the Marshall Hall Conference" (D. Jungnickel and S. A. Vanstone, editors), Wiley, pp. 263–273. The 56.15% Upper Bound on Turán Densities: Rahil Baber (2012), "Turán Densities of Hypercubes", https://arxiv.org/abs/1201.3587 The 5/9 construction of acute triangles: https://math.stackexchange.com/questions/2892862/find-the-maximum-percentage-of-acute-angled-triangles-in-a-plane-with-100-poin/ I thank Rahil Baber for many useful discussions: https://www.rahilbaber.com