From DNF compression to sunflower theorems via regularity

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 the DNF compression result. The main approach is based on regularity of set systems and a structure-vs-pseudorandomness approach to the sunflower conjecture.

[1]  Xin Li,et al.  Sunflowers and Quasi-sunflowers from Randomness Extractors , 2018, Electron. Colloquium Comput. Complex..

[2]  Omer Reingold,et al.  DNF sparsification and a faster deterministic counting algorithm , 2012, 2012 IEEE 27th Conference on Computational Complexity.

[3]  Shachar Lovett,et al.  Rectangles Are Nonnegative Juntas , 2015, SIAM J. Comput..

[4]  Benjamin Rossman.  The Monotone Complexity of k-clique on Random Graphs , 2010, FOCS.

[5]  Alexandr V. Kostochka,et al.  Extremal Problems on Δ-Systems , 2000 .

[6]  Noga Alon,et al.  On sunflowers and matrix multiplication , 2012, 2012 IEEE 27th Conference on Computational Complexity.

[7]  H. L. Abbott,et al.  Intersection Theorems for Systems of Sets , 1972, J. Comb. Theory, Ser. A.

[8]  Shachar Lovett,et al.  DNF sparsification beyond sunflowers , 2018, Electron. Colloquium Comput. Complex..