This article deals with the Maximum Weight Stable Set (MWS) problem (and some other related NP-hard problems) and the class of P₆-free graphs. The complexity status of MWS is open for P₆-free graphs and is open even for P₅-free graphs (as a long standing open problem). Several results are known for MWS on subclasses of P₅-free: in particular, MWS can be solved for k-colorable P₅-free graphs in polynomial time for every k (depending on k) and more generally for (P₅,K_p)-free graphs (depending on p), which is a useful result since for every graph G one can easily compute a k-coloring of G, with k not necessarily minimum. This article studies the MWS problem for k-colorable P₆-free graphs and more generally for (P₆,K_p)-free graphs. Though we were not able to define a polynomial time algorithm for this problem for every k, this article introduces: (i) some structure properties of P₆-free graphs with respect to stable sets, (ii) two reductions for MWS on (P₆,K_p)-free graphs for every p, (iii) three polynomial time algorithms to solve MWS respectively for 3-colorable P₆-free, for 4-colorable P₆-free, and for (P₆,K₄)-free graphs (the latter allows one to state, together with other known results, that MWS can be solved for (P₆,F)-free graphs in polynomial time where F is any four vertex graph).