Sunflowers and Quasi-sunflowers from Randomness Extractors

The Erdős-Rado sunflower theorem (Journal of Lond. Math. Soc. 1960) is a fundamental result in combinatorics, and the corresponding sunflower conjecture is a central open problem. Motivated by applications in complexity theory, Rossman (FOCS 2010) extended the result to quasi-sunflowers, where similar conjectures emerge about the optimal parameters for which it holds. In this work, we exhibit a surprising connection between the existence of sunflowers and quasisunflowers in large enough set systems, and the problem of constructing certain randomness extractors. This allows us to re-derive the known results in a systemic manner, and to reduce the relevant conjectures to the problem of obtaining improved constructions of the randomness extractors.

[1]  G. Szekeres,et al.  A combinatorial problem in geometry , 2009 .

[2]  Oded Goldreich,et al.  Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity , 1988, SIAM J. Comput..

[3]  Omer Reingold,et al.  DNF Sparsification and a Faster Deterministic Counting Algorithm , 2012, Computational Complexity Conference.

[4]  Xin Li,et al.  Three-Source Extractors for Polylogarithmic Min-Entropy , 2015, 2015 IEEE 56th Annual Symposium on Foundations of Computer Science.

[5]  E. Szemerédi On sets of integers containing k elements in arithmetic progression , 1975 .

[6]  P. Erdös Some remarks on the theory of graphs , 1947 .

[7]  Amnon Ta-Shma,et al.  Explicit two-source extractors for near-logarithmic min-entropy , 2016, Electron. Colloquium Comput. Complex..

[8]  E. Szemerédi Regular Partitions of Graphs , 1975 .

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

[10]  D. Zuckerman,et al.  Explicit two-source extractors and resilient functions , 2016, Electron. Colloquium Comput. Complex..

[11]  Guy Kindler,et al.  Simulating independence: new constructions of condensers, ramsey graphs, dispersers, and extractors , 2005, STOC '05.

[12]  Benjamin Rossman,et al.  The Monotone Complexity of k-clique on Random Graphs , 2010, 2010 IEEE 51st Annual Symposium on Foundations of Computer Science.

[13]  Prasad Raghavendra,et al.  Approximating rectangles by juntas and weakly-exponential lower bounds for LP relaxations of CSPs , 2016, STOC.

[14]  T. Tao,et al.  The primes contain arbitrarily long arithmetic progressions , 2004, math/0404188.

[15]  J. Spencer Intersection Theorems for Systems of Sets , 1977, Canadian Mathematical Bulletin.

[16]  Oded Goldreich,et al.  Unbiased bits from sources of weak randomness and probabilistic communication complexity , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[17]  Gil Cohen,et al.  Two-Source Extractors for Quasi-Logarithmic Min-Entropy and Improved Privacy Amplification Protocols , 2016, Electron. Colloquium Comput. Complex..

[18]  Xin Li,et al.  Improved non-malleable extractors, non-malleable codes and independent source extractors , 2016, Electron. Colloquium Comput. Complex..

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

[20]  Omer Reingold,et al.  DNF Sparsification and a Faster Deterministic Counting , 2012, Electron. Colloquium Comput. Complex..