Partitioning dense uniform hypergraphs

Abstract

Let $$r\ge 3$$ and $$k\ge 2$$ be fixed integers, and let H be an r-uniform hypergraph with n vertices and m edges. In 1997, Bollobas and Scott conjectured that H has a vertex-partition into k sets with at most $$m/k^r+o(m)$$ edges in each set. So far, this conjecture was confirmed when $$r=3$$ or $$m=\Omega (n^{r-1+o(1)})$$ . In this paper, we show that it holds for $$m=\Omega (n^{r-3+\epsilon })$$ for any $$\epsilon >0$$ .