A quantitative improvement on the hypergraph Balog-Szemer\'{e}di-Gowers theorem

TL;DR

This paper presents a quantitative enhancement of the hypergraph version of the Balog-Szemerédi-Gowers theorem and establishes an 'almost all' variant, advancing the understanding of hypergraph combinatorics.

Contribution

It provides a new quantitative bound and proves an 'almost all' version of the hypergraph Balog-Szemerédi-Gowers theorem, extending prior results.

Findings

Improved quantitative bounds for the hypergraph BSG theorem

Proved the hypergraph 'almost all' BSG theorem

Enhanced understanding of hypergraph combinatorial structures

Abstract

In this note, we obtain a quantitative improvement on the hypergraph variant of the Balog-Szemer\'{e}di-Gowers theorem due to Sudakov, Szemer\'{e}di, and Vu [Duke Math. J.129.1 (2005): 129--155]. Additionally, we prove the hypergraph variant of the ``almost all'' version of Balog-Szemer\'{e}di-Gowers theorem.