The size multipartite Ramsey numbers of paths with respect to complete balanced multipartite graphs (Q2910580)

Let \(K_{j \times t}\) denote the complete multipartite graph with \(j\) partite sets, each of \(t\) vertices. The author continues an investigation begun in earlier papers [``On size multipartite Ramsey numbers for paths versus cycles of three or four vertices,'' Far East J.\ Appl.\ Math.\ 44, No.\ 2, 109--116 (2010; Zbl 1225.05180); ``On the size multipartite Ramsey numbers for small path versus cocktail party graphs,'' Far East J.\ Appl.\ Math.\ 55, No.\ 1, 53--60 (2011; Zbl 1232.05134)] where the concept of \textit{size multipartite Ramsey number} [\textit{A. P. Burger} and \textit{J. H. van Vuuren}, ``Ramsey numbers in complete balanced multipartite graphs.\ II: Size numbers,'' Discrete Math.\ 283, No.\ 1--3, 45--49 (2004; Zbl 1042.05065)] is generalized. For given graphs \(G_1\), \(G_2\), the \textit{(generalized) size multipartite Ramsey number} \(m_j\left(G_1,G_2\right)\) is the smallest integer \(t\) such that every factorization of graph \(K_{j\times t}:=F_1\oplus F_2\) (where the edges have been partitioned into \(E\left(F_1\right)\) and \(E\left(F_2\right)\)) satisfies the following condition: either \(F_1\) contains \(G_1\) as a subgraph, or \(F_2\) contains \(G_2\) as a subgraph. \textbf{Theorem 1.} For integer \(n\geq2\), NEWLINENEWLINE\[NEWLINEm_3\left(P_n,K_{2\times2}\right) = NEWLINE\begin{cases}NEWLINE2&\text{for \(n=3\)},\\ NEWLINE3&\text{for \(4\leq n\leq 6\)},\\ NEWLINE\left\lceil\frac{n+1}3\right\rceil&\text{for \(n\geq7\)}.NEWLINE\end{cases}NEWLINE\]