Constructions in Ramsey theory

TL;DR

This paper advances Ramsey theory by providing new constructions, establishing superexponential lower bounds for hypergraph Ramsey numbers, and improving bounds on the Erd ext{"o}s-Rogers function, thus deepening understanding of hypergraph combinatorics.

Contribution

It introduces novel constructions and bounds for hypergraph Ramsey numbers and the Erd ext{"o}s-Rogers function, improving upon previous exponential and logarithmic bounds.

Findings

Proved superexponential lower bounds for $r_4(5,n)$ and $r_k(k+1,n)$.

Established an iterated logarithm upper bound for the hypergraph Erd ext{"o}s-Rogers function.

Generalized Erd ext{"o}s-Hajnal results to $k$-uniform hypergraphs.

Abstract

We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform Ramsey number $r_4(5,n)$, and the same for the iterated $(k-4)$-fold logarithm of the $k$-uniform version $r_k(k+1,n)$. This is the first improvement of the original exponential lower bound for $r_4(5,n)$ implicit in work of Erd\H os and Hajnal from 1972 and also improves the current best known bounds for larger $k$ due to the authors. Second, we prove an upper bound for the hypergraph Erd\H os-Rogers function $f^k_{k+1, k+2}(N)$ that is an iterated $(k-13)$-fold logarithm in $N$. This improves the previous upper bounds that were only logarithmic and addresses a question of Dudek and the first author that was reiterated by Conlon, Fox and Sudakov. Third, we generalize the results of Erd\H os and Hajnal about the 3-uniform Ramsey number of $K_4$ minus an edge versus a clique to $k$-uniform hypergraphs.