For every fixed integer k≥ 2, there exists a constant C_k>0 such that every 3-uniform hypergraph H with m edges has a partition V(H)=V_1 sqcup…sqcup V_k satisfying e bigl(H[V_i]bigr)≤ fracmk^3+C_k m^2/3 for every i∈1,…,k, where H[V_i] is the subhypergraph induced by V_i.

Scott’s conjecture on judicious partitions of 3-uniform hypergraphs

Scott’s conjecture predicts sharp judicious partitions of uniform hypergraphs. For the three-uniform case, the latest preprint claims the conjectured error scale, improving the previous bound.

Known results

Bollobás and Scott (2000) conjectured that every $r$ -uniform hypergraph with $m$ edges admits an $r$ -partition in which every class meets at least $\frac{rm}{2r-1}$ edges.
For $r=3$ , Bollobás and Scott proved the weaker bound $\frac{5m-1}{9}$ .
Halsegrave and Ma–Yu obtained successive improvements to $0.6m$ and $0.65m+o(m)$ .
Lin and collaborators (2018) proved the three-uniform bound $\frac{19}{27}m-O(m^{6/7})$ , sharp up to the error term.

October 2026 claimed improvement

Siwei Lin and Qinghou Zeng’s preprint claims an $O(m^{2/3})$ error term, matching the conjectured scale and improving $O(m^{6/7})$ . This would settle the sharp-order error term for the three-uniform case, but the claim has not been independently verified in the retrieved sources.

Current status (as of October 2026): The three-uniform sharp-order error term is claimed solved but remains unverified; analogous conjectures for higher uniformities remain open.

Scott’s conjecture on judicious partitions of 3-uniform hypergraphs

Primary source

Additional references

Progress summary

Solutions 0