One more Tur\'an number and Ramsey number for the loose 3-uniform path of length three

TL;DR

This paper refines the analysis of Turán numbers for a specific 3-uniform hypergraph path, leading to the exact determination of its 10-color Ramsey number by computing a higher-order Turán number.

Contribution

It introduces the fifth order Turán number for the path and applies it to precisely determine the Ramsey number for 10 colors.

Findings

Computed the fifth order Turán number for the path P.

Confirmed the Ramsey number R(P;10)=16.

Enhanced understanding of Turán and Ramsey numbers for 3-uniform hypergraphs.

Abstract

Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-color Ramsey number for $P$ is $R(P;r)=r+6$ for $r\le 9$. The proof of this result relies on a careful analysis of the Tur\'an numbers for $P$. In this paper, we refine this analysis further and compute the fifth order Tur\'an number for $P$, for all $n$. Using this number for $n=16$, we confirm the formula $R(P;10)=16$.