Asaf Shapira - Local vs Global Combinatorics

Explore the fascinating world of extremal combinatorics in this 46-minute lecture by Asaf Shapira, presented by the International Mathematical Union. Delve into key concepts such as the Brown-Erdos-Sós Conjecture, hypergraphs without grids, and removal lemmas. Examine the interplay between local and global combinatorial properties, investigate tight bounds for hypergraph regularity, and discover variants of removal lemmas including infinite and polynomial-bounded versions. Gain insights into generalized Turán problems and the application of combinatorial techniques to solve complex mathematical challenges.

Intro
What is Extremal Combinatorics All About?
Guiding Problem 2
The Brown-Erdos-Sós Conjecture
Approximate Brown-Erdős-Sós Conjecture
Two Competing Conjectures
Hypergraphs without Grids
Deriving the Removal Lemma from Regularity Lemma
Bounds for Removal Lemma
The Hypergraph Removal Lemma
A Tight Lower Bound for Hypergraph Regularity
A Tight Upper Bound for Regularity with Modificatio
Almost Monochromatic Sets
Variants of the Removal Lemma
An Infinite Removal Lemma
Removal Lemmas with Polynomial Bounds
Removal Lemmas with Prescribed Growth
A Generalized Turán Problem