Independent sets in hypergraphs with a forbidden link

TL;DR

This paper constructs a 3-uniform hypergraph with minimal independence number and limited edges among four vertices, solving a longstanding problem in Ramsey theory and extending bounds to other hypergraph Ramsey numbers.

Contribution

It provides a probabilistic construction of hypergraphs with optimal independence number and edge restrictions, resolving an open problem in hypergraph Ramsey theory.

Findings

Constructed hypergraph with independence number O(log N / log log N)

Proved the bound is tight, solving a longstanding open problem

Extended results to various hypergraph Ramsey numbers

Abstract

We give a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding open problem of Erd\H{o}s and Hajnal in Ramsey theory. We further extend this result to prove tight bounds on various other hypergraph Ramsey numbers.