For given graphs G and H, the Ramsey number R(G,H) is the smallest positive integer N such that for every graph F of order N the following holds: either F contains G as a subgraph or the complement of F contains H as a subgraph. For the path Pn and the Jahangir graph Jm, it is proved that R(P4, J4) = 6, R(Pn, J4) = n + 1 for n ≥ 5 and R(Pn, Jm) = n + m 2 − 1 for even m ≥ 6 and n ≥ (2m − 1)(m2 − 1) + 1. We also suggest an open problem. Mathematics Subject Classifications: 05C55, 05D10