Judicious partitions of uniform hypergraphs

The vertices of any graph with m edges may be partitioned into two parts so that each part meets at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tfrac{{2m}} {3}$$\end{document} edges. Bollobás and Thomason conjectured that the vertices of any r-uniform hypergraph with m edges may likewise be partitioned into r classes such that each part meets at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tfrac{r} {{2r - 1}}$$\end{document} edges. In this paper we prove the weaker statement that, for each r ≥ 4, a partition into r classes may be found in which each class meets at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tfrac{r} {{3r - 4}}$$\end{document} edges, a substantial improvement on previous bounds.