For given k-uniform hypergraphs \({\mathcal {G}}\) and \({\mathcal {H}}\) , the Ramsey number \(R({\mathcal {G}},{\mathcal {H}})\) is the smallest positive integer N such that in every red-blue coloring of the edges of the complete k-uniform hypergraph on n vertices there is either a red copy of \({\mathcal {G}}\) or a blue copy of \({\mathcal {H}}\) . In this paper, results are given which permit the \(R(m{\mathcal {G}},n{\mathcal {H}})\) to be evaluated exactly when m or n is large and \({\mathcal {G}}\) is a k-uniform hypergraph with the maximum independent set that intersects each edge in \(k-1\) vertices and \({\mathcal {H}}\) is a k-uniform hypergraph with a vertex so that the hypergraph induced by the edges containing this vertex is a star. There are several examples for such \({\mathcal {G}}\) and \({\mathcal {H}}\) , among them are any disjoint union of k-uniform hypergraphs involving loose paths, loose cycles, tight paths, tight cycles, stars, Kneser hypergraphs and complete k-uniform k-partite hypergraphs for \({\mathcal {G}}\) and linear hypergraphs for \({\mathcal {H}}\) . As an application, \(R(m{\mathcal {G}},n{\mathcal {H}})\) is determined when m or n is large and \({\mathcal {G}}\) , \({\mathcal {H}}\) are either loose paths, loose cycles, tight paths or stars. Moreover, for given k-uniform hypergraphs \({\mathcal {G}}\) and \({\mathcal {H}}\) and positive integers m, n, some bounds are given for \(R(m{\mathcal {G}},n{\mathcal {H}})\) which enable us to compute \(R(m{\mathcal {G}},n{\mathcal {H}})\) when \(m\ge n\ge 1\) and \({\mathcal {G}}, {\mathcal {H}}\) are either 3-uniform loose path \({\mathcal {P}}_r^3\) or loose cycle \({\mathcal {C}}_r^3\) : We shall show that for every \(m\ge n\ge 1\) and \(r\ge s\) , \(R(m{\mathcal {C}}_r^3,n{\mathcal {C}}_s^3)=2rm+\Big \lfloor \frac{s+1}{2}\Big \rfloor n-1,\) and \(R(m{\mathcal {P}}_r^3,n{\mathcal {P}}_s^3)=(2r+1)m+\Big \lfloor \frac{s+1}{2}\Big \rfloor n-1.\)
