Ramsey Number of Hypergraph Paths

Abstract

Let $H = (V, E)$ be a $k$-uniform hypergraph. For $1 ≤ s ≤ k − 1,$ an $s$-path $P^{(k,s)}_n$ of length $n$ in $H$ is a sequence of distinct vertices $v_1, v_2, · · · , v_{s+n(k−s)}$ such that $\{v_{1+i(k-s)}, \cdots, v_{s+(i+1)(k-s)}\}$ is an edge of $H$ for each $0 ≤ i ≤ n−1.$ In this paper, we prove that $R(P^ {(3s,s)}_n , P^{(3s,s)}_3) = (2n + 1)s + 1$ for $n ≥ 3.$