Lower bounds of the size multipartite Ramsey numbers m_j(P_n,K_j×b)

For given two graphs G₁ and G₂, the size Ramsey multipartite numbers m_j(G₁,G₂) is the smallest integer t such that every factorization of graph K_j×t := F₁ ⊕ F₂ satisfies the following condition: either F₁ contains G₁ as a subgraph or F₂ contains G₂ as a subgraph. In this paper, we derive some lower bound for the size multipartite Ramsey numbers m_j(P_n,K_j×b) with integers j,n ≥ 3 and b ≥ 2.