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: $R({ \mathcal P}^{ r,r/2}_n,{ \mathcal P}^{ r,r/2}_3)=R({ \mathcal P}^{ r,r/2}_n,{ \mathcal P}^{ r,r/2}_4)=\tfrac{ (n+1)r}{ 2}+1.$
Categories: math.CO