Let　r　≥　3　and　k　≥　2　be　fixed　integers.　Bollobás　and　Scott　conjectured　that　every　r-uniform　hypergraph　with　m　edges　has　a　vertex　partition　into　k　sets　with　at　most　m/kr+o(m)　edges　in　each　set,　and　proved　the　conjecture　in　the　case　r　=　3.　In　this　paper,　we　confirm　this　conjecture　in　the　case　r　=　4　by　showing　that　every　4-uniform　hypergraph　with　m　edges　has　a　vertex　partition　into　k　sets　with　at　most　m/k4　+　O(m8/9)　edges　in　each　set.　©　2018　Society　for　Industrial　and　Applied　Mathematics.