Three Proofs of the Hypergraph Ramsey Theorem (An exposition)

TL;DR

This paper presents three different proofs of the 3-hypergraph Ramsey Theorem, compares their bounds, and provides explicit bounds for various cases, offering insights into hypergraph Ramsey numbers.

Contribution

It compiles and analyzes multiple proofs of the hypergraph Ramsey Theorem, improving bounds and clarifying the relationships between different approaches.

Findings

Explicit bounds for 2-color and c-color cases

Improved bounds on hypergraph Ramsey numbers

Detailed analysis of Conlon-Fox-Sudakov construction

Abstract

Ramsey, Erdos-Rado, and Conlon-Fox-Sudakov have given proofs of the 3-hypergraph Ramsey Theorem with better and better upper bounds on the 3-hypergraph Ramsey Number. Ramsey and Erdos-Rado also prove the a-hypergraph Ramsey Theorem. Conlon-Fox-Sudakov note that their upper bounds on the 3-hypergraph Ramsey Numbers, together with a recurrence of Erdos-Rado (which was the key to the Erdos-Rado proof), yield improved bounds on the a-hypergraph Ramsey numbers. We present all of these proofs and state explicit bounds for the 2-color case and the c-color case. We give a more detailed analysis of the construction of Conlon-Fox-Sudakov and hence obtain a slightly better bound.