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Sunflowers and Quasi-Sunflowers from Randomness Extractors

Authors Xin Li, Shachar Lovett, Jiapeng Zhang

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Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • Sunflower conjecture
  • Quasi-sunflowers
  • Randomness Extractors

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Abstract

The Erdös-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 quasi-sunflowers in large enough set systems, and the problem of constructing (or existing) certain randomness extractors. This allows us to re-derive the known results in a systematic manner, and to reduce the relevant conjectures to the problem of obtaining improved constructions of the randomness extractors.

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Xin Li, Shachar Lovett, and Jiapeng Zhang. Sunflowers and Quasi-Sunflowers from Randomness Extractors. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 116, pp. 51:1-51:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2018.51

Author Details

Xin Li
  • Johns Hopkins University, Baltimore, USA
Shachar Lovett
  • University of California San Diego, La Jolla, USA
Jiapeng Zhang
  • University of California San Diego, La Jolla, USA

Funding

  • Li, Xin: Research supported by NSF award CCF-1617713
  • Lovett, Shachar: Research supported by NSF CCF-1614023
  • Zhang, Jiapeng: Research supported by NSF CCF-1614023

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