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):
some.3b1b.co/
My video won an Honorable Mention!

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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):
   • "A cute" triangle problem from the 1970 IM...  

Proof of Mantel's theorem (via counting subgraphs) by J. A. Bondy (1997):
www.sciencedirect.com/science/article/pii/S0012365…

Probabilistic proof of Turán's theorem in graph theory:
en.wikipedia.org/wiki/Tur%C3%A1n%27s_theorem#Proba…
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:
en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Stone_the…

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", arxiv.org/abs/1201.3587

The 5/9 construction of acute triangles:
math.stackexchange.com/questions/2892862/find-the-…

I thank Rahil Baber for many useful discussions:
www.rahilbaber.com/