While there is a full π01-class of nonrecursive sets where no set is pdominant, there is no π01-class containing only pdominant sets.  ...  No set which is low for Martin-Löf random is pdominant. There is a low r.e. set which is pdominant and a high r.e. set which is not pdominant.  ...  Acknowledgments The authors want to thank Noam Greenberg for discussions and correspondence on this subject; furthermore they are greatful to the anonymous referees for detailed and comprehensive comments  ...