Abstract: The problem of determining the independence number
of (hyper)graphs has tight connections to questions in discrete geometry,
coding theory, number theory, theoretical computer science and combinatorics.
One of the most famous early examples is the result of Komlos-Pintz-Szemeredi
from 1982 on the independence number of 3-uniform hypergraphs which made
important progress on the decades old Heilbronn problem. I will begin by
explaining this result and some of these connections. I will then describe
recent work in this area which shows that hypergraphs have a significantly
different behavior than graphs when it comes to independent sets. This answers
a question posed by Ajtai-Erdos-Komlos-Szemeredi (1981), and disproves conjectures of
deCaen (1986), Frieze and the speaker (2007), and several others.