We are indebted to Spink and Tiba Spink and Tiba [3] for pointing out that the key technical lemma (Lemma 4) is incorrect in our paper [2]. We tried to make a revision for fixing the gap. However, with the help of mathematical software Lingo, we found that our lemma has counterexamples and its conclusion is incorrect. The Bollobás--Scott conjecture Spink and Tiba [3], “Every $r$-uniform hypergraph with $m$ edges has a vertex partition into $k$ sets with at most $m/k^r+o(m)$ edges in each set,” remains open for $r\ge4$ and seems difficult. The following result shows that Lemma 4 in [2] is wrong even in the case $k=2$.

Let $r\ge 3$ and $k\ge 2$ 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 $m/k^r+o(m)$ 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 $m/k^4+O(m^{8/9})$ edges in each set.