Judicious partitions of 3-uniform hypergraphs

The vertices of any graph with $m$ edges can be partitioned into two parts so that each part meets at least $\frac{2m}{3}$ edges. Bollob\'as and Thomason conjectured that the vertices of any $r$-uniform graph may be likewise partitioned into $r$ classes such that each part meets at least $cm$ edges, with $c=\frac{r}{2r-1}$. In this paper, we prove this conjecture for the case $r=3$. In the course of the proof we shall also prove an extension of the graph case which was conjectured by Bollob\'as and Scott.