The Ramsey number for 4-uniform tight cycles

TL;DR

This paper determines the asymptotic Ramsey number for 4-uniform tight cycles and paths, establishing precise growth rates and confirming their tightness, advancing understanding in hypergraph Ramsey theory.

Contribution

It proves the asymptotic value of the Ramsey number for 4-uniform tight cycles and paths, providing new bounds and confirming their tightness.

Findings

Ramsey number for 4-uniform tight cycle is (5 + o(1))n.

Ramsey number for 4-uniform tight path is (5/4 + o(1))n.

Results are asymptotically tight.

Abstract

A $k$-uniform tight cycle is a $k$-graph with a cyclic ordering of its vertices such that its edges are precisely the sets of $k$ consecutive vertices in that ordering. A $k$-uniform tight path is a $k$-graph obtained by deleting a vertex from a $k$-uniform tight cycle. We prove that the Ramsey number for the $4$-uniform tight cycle on $4n$ vertices is $(5 +o(1))n$. This is asymptotically tight. This result also implies that the Ramsey number for the $4$-uniform tight path on $n$ vertices is $(5/4 + o(1))n$.