On tight tree-complete hypergraph Ramsey numbers

TL;DR

This paper investigates the growth of hypergraph Ramsey numbers for tight trees, establishing polynomial bounds for 3- and 4-uniform cases, and introduces new constructions inspired by prior work.

Contribution

It provides the first polynomial bounds for 4-uniform tight trees' Ramsey numbers, extending known results from 3-uniform cases with novel constructions.

Findings

For 3-uniform tight trees, R(T,n)=Θ(n^2).

For 4-uniform tight trees, R(T,n)=Θ(n^3).

Introduces new construction methods inspired by Cooper and Mubayi.

Abstract

Chv\'atal showed that for any tree $T$ with $k$ edges the Ramsey number $R(T,n)=k(n-1)+1$ ("Tree-complete graph Ramsey numbers." Journal of Graph Theory 1.1 (1977): 93-93). For $r=3$ or $4$, we show that, if $T$ is an $r$-uniform non-trivial tight tree, then the hypergraph Ramsey number $R(T,n)=\Theta(n^{r-1})$. The 3-uniform result comes from observing a construction of Cooper and Mubayi. The main contribution of this paper is the 4-uniform construction, which is inspired by the Cooper-Mubayi 3-uniform construction.