Ramsey numbers for the path with three edges (Q1084414)

Let k be a natural number and let \(P_ 3\) denote the path with three edges. The Ramsey number \(r(P_ 3,k)\) is the minimum number of vertices in a complete graph for which every k-coloring admits a monochromatic copy of \(P_ 3\). The Ramsey number \(r(F_ 3,k)\) is the minimum number of vertices in a complete graph for which every k-coloring admits a monochromatic connected subgraph or more than three vertices. This note contains the following two \(results:\) r(P\({}_ 3,k)=2k+2\), if \(k\equiv (mod 3)\), \(r(P_ 3,k)=2k+1\), if \(k\equiv 0\) or 2(mod 3), \(k\neq 3\), \(r(P_ 3,k)=6\), if \(k=3,\) r(F\({}_ 3,k)=2k+2\), if \(k\equiv 1(mod 3)\), \(r(F_ 3,k)=2k\), if \(k=3\) or \(k\equiv 2(mod 3)\), \(r(F_ 3,k)=2k+1\) if \(k=3^ mh>3\), \(h\in 1(mod 3)\).