Calculating Ramsey numbers by partitioning coloured graphs

In this paper we prove a new result about partitioning coloured complete graphs and use it to determine certain Ramsey numbers exactly. The partitioning theorem we prove is that for k at least 1, in every edge colouring of a complete graph with the colours red and blue, it is possible to cover all the vertices with k disjoint red paths and a disjoint blue balanced complete (k+1)-partite graph. When the colouring is connected in red, we prove a stronger result - that it is possible to cover all the vertices with k red paths and a blue balanced complete (k+2)-partite graph. Using these results we determine the Ramsey number of a path on n vertices, versus a balanced complete k-partite graph, with m vertices in each part, whenever m-1 is divisible by n-1. This generalizes a result of Erdos who proved the m=1 case of this result. We also determine the Ramsey number of a path on n vertices versus the power of a path on n vertices. This solves a conjecture of Allen, Brightwell, and Skokan.