The Ramsey Number of 3-Uniform Loose Path Versus Star

Abstract

For given 3-uniform hypergraphs $\mathcal {H}$ and $\mathcal {G}$, the Ramsey number $R(\mathcal {H}, \mathcal {G})$ is the smallest number N such that each red-blue coloring of the edges of the complete 3-uniform hypergraph $\mathcal {K}_N^3$ contains either a red copy of $\mathcal {H}$ or a blue copy of $\mathcal {G}$. Let $\mathcal {C}_n^3$, $\mathcal {P}_n^3$ and $\mathcal {S}_n^3$ denote the 3-uniform loose cycle, loose path and star with n edges, respectively. The 2-color Ramsey number of 3-uniform loose path and loose cycle has been determined completely by several researchers. In this paper, we prove that $R(\mathcal {P}_m^3, \mathcal {S}_n^3)=2n+1+\left\lfloor \frac{m-1}{2}\right\rfloor$ for all m, n with $n\ge 2m\ge 2$ and $R(\mathcal {P}_m^3, \mathcal {S}_n^3)=2m+1$ for all m, n with $m\ge \left\lfloor \frac{9}{4}n\right\rfloor$.