From DNF compression to sunflower theorems via regularity

TL;DR

This paper establishes a connection between improved DNF compression bounds and progress on the sunflower conjecture using regularity and structure-vs-pseudorandomness techniques.

Contribution

It demonstrates that advances in DNF compression can lead to better bounds for the sunflower conjecture, reversing previous implications.

Findings

Improved DNF compression bounds imply better sunflower conjecture bounds.

Regularity methods are effective in analyzing set systems.

Structure-vs-pseudorandomness approach advances understanding of sunflower structures.

Abstract

The sunflower conjecture is one of the most well-known open problems in combinatorics. It has several applications in theoretical computer science, one of which is DNF compression, due to Gopalan, Meka and Reingold [Computational Complexity 2013]. In this paper, we show that improved bounds for DNF compression imply improved bounds for the sunflower conjecture, which is the reverse direction of [Computational Complexity 2013]. The main approach is based on regularity of set systems and a structure-vs-pseudorandomness approach to the sunflower conjecture.