Refined Tur\'an numbers and Ramsey numbers 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, confirming known Ramsey numbers for certain cases and extending the understanding of Turán numbers.

Contribution

It computes the third and fourth order Turán numbers for the loose 3-uniform path of length three, extending previous results and confirming Ramsey numbers for additional cases.

Findings

Computed third and fourth order Turán numbers for the hypergraph path

Confirmed Ramsey number formula for r=8,9

Extended Turán number analysis for hypergraph paths

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 $r\le 7$. 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, for all $n$, the third and fourth order Tur\'an numbers for $P$. With the help of the former, we confirm the formula $R(P;r)=r+6$ for $r\in\{8,9\}$.