The independent neighborhoods process

TL;DR

This paper proves that a random greedy process on hypergraphs produces a structure with small independence number, leading to bounds on hypergraph Ramsey numbers and extending classical results to higher uniformities.

Contribution

It establishes a new bound on the independence number in hypergraphs generated by a random greedy process, generalizing classical combinatorial results.

Findings

The process terminates with hypergraphs having independence number $O((n \, \log n)^{1/r})$.

The hypergraph Ramsey number $r(T^{(r)}, K_s^{(r)})$ is of order $s^r/\log s$.

Answers to open questions in hypergraph Ramsey theory.

Abstract

A triangle $T^{(r)}$ in an $r$-uniform hypergraph is a set of $r+1$ edges such that $r$ of them share a common $(r-1)$-set of vertices and the last edge contains the remaining vertex from each of the first $r$ edges. Our main result is that the random greedy triangle-free process on $n$ points terminates in an $r$-uniform hypergraph with independence number $O((n \log n)^{1/r})$. As a consequence, using recent results on independent sets in hypergraphs, the Ramsey number $r(T^{(r)}, K_s^{(r)})$ has order of magnitude $s^r/\log s$. This answers questions posed in~\cite{BFM, KMV} and generalizes the celebrated results of Ajtai-Koml\'os-Szemer\'edi~\cite{AKS} and Kim~\cite{K} to hypergraphs.