The Ramsey number of generalized loose paths in uniform Hypergrpahs

TL;DR

This paper determines the exact Ramsey number for certain even-uniform hypergraph paths, specifically for $r/2$-paths, extending recent work on 1-paths in hypergraphs.

Contribution

It provides the first exact value for the Ramsey number of $r/2$-paths in even uniform hypergraphs, filling a gap in hypergraph Ramsey theory.

Findings

Exact Ramsey number for ${ m P}^{r,r/2}_n$ and ${ m P}^{r,r/2}_3$ is $rac{(n+1)r}{2}+1$.

Established the same Ramsey number for ${ m P}^{r,r/2}_n$ and ${ m P}^{r,r/2}_4$.

Extended understanding of hypergraph path Ramsey numbers for even uniformity.

Abstract

Let $H=(V,E)$ be an $r$-uniform hypergraph. For each $1 \leq s \leq r-1$, an $s$-path ${\mathcal P}^{r,s}_n$ of length $n$ in $H$ is a sequence of distinct vertices $v_1,v_2,\ldots,v_{s+n(r-s)}$ such that $\{v_{1+i(r-s)},\ldots, v_{s+(i+1)(r-s)}\}\in E(H)$ for each $0 \leq i \leq n-1$.Recently, the Ramsey number of $1$-paths in uniform hypergraphs has received a lot of attention. In this paper, we consider the Ramsey number of $r/2-$paths for even $r$. Namely, we prove the following exact result: $R({\mathcal P}^{r,r/2}_n,{\mathcal P}^{r,r/2}_3)=R({\mathcal P}^{r,r/2}_n,{\mathcal P}^{r,r/2}_4)=\tfrac{(n+1)r}{2}+1.$