The Ramsey number of generalized loose paths in uniform Hypergrpahs

Abstract

Let $H=(V,E)$ be an $r$-uniform hypergraph. For each $1\le s\le r-1$, an $s$-path ${\mathcal{P}}_n^{r,s}$ of length $n$ in $H$ is a sequence of distinct vertices $v_1,v_2,\dots ,v_{s+n(r-s)}$ such that $\{v_{1+i(r-s)},\dots ,v_{s+(i+1)(r-s)}\}\in E(H)$ for each $0\le i\le n-1$.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: $R({\mathcal{P}}_n^{r,r/2},{\mathcal{P}}_3^{r,r/2})=R({\mathcal{P}}_n^{r,r/2},{\mathcal{P}}_4^{r,r/2})=\frac{(n+1)r}{2}+1.$

Journal Ref: Discrete Math

Cited by in corpus (2)

• Regular Uniform Hypergraphs, $s$-Cycles, $s$-Paths and Their largest Laplacian H-Eigenvalues
• Ordered Ramsey numbers of loose paths and matchings