Judiciously 3-partitioning 3-uniform hypergraphs

TL;DR

This paper improves the known bounds on partitioning vertices of 3-uniform hypergraphs to meet a significant fraction of edges, advancing understanding of hypergraph partitions and confirming a special case of a longstanding conjecture.

Contribution

It establishes a new asymptotic bound of 19/27 m + o(m) for vertex partitions in 3-uniform hypergraphs, resolving a specific case of Bollobás and Scott's conjecture.

Findings

Improved the lower bound to 19/27 m + o(m)

Bound is asymptotically best possible

Resolved a special case of a conjecture by Bollobás and Scott

Abstract

Bollob\'as, Reed and Thomason proved every $3$-uniform hypergraph $\mathcal{H}$ with $m$ edges has a vertex-partition $V(\mathcal{H})=V_1 \sqcup V_2 \sqcup V_3$ such that each part meets at least $\frac{1}{3}(1-\frac{1}{e})m$ edges, later improved to $0.6m$ by Halsegrave and improved asymptotically to $0.65m+o(m)$ by Ma and Yu. We improve this asymptotic bound to $\frac{19}{27}m+o(m)$, which is best possible up to the error term, resolving a special case of a conjecture of Bollob\'as and Scott.