Improved bounds for the sunflower lemma

@article{Alweiss2019ImprovedBF,
  title={Improved bounds for the sunflower lemma},
  author={Ryan Alweiss and Shachar Lovett and Kewen Wu and Jiapeng Zhang},
  journal={Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing},
  year={2019}
}
A sunflower with r petals is a collection of r sets so that the intersection of each pair is equal to the intersection of all. Erdős and Rado proved the sunflower lemma: for any fixed r, any family of sets of size w, with at least about w w sets, must contain a sunflower. The famous sunflower conjecture is that the bound on the number of sets can be improved to c w for some constant c. In this paper, we improve the bound to about (logw) w . In fact, we prove the result for a robust notion of… 
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This paper gives a simplified proof of deterministic lifting (in both the tree-like and dag-like settings) and conjecture that it can be further improved to polylogarithmic, similar to the known bounds for the corresponding robust sunflower lemmas.
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We present some problems and results about variants of sunflowers in families of sets. In particular, we improve an upper bound of the first author, Korner and Monti on the maximum number of binary
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Coding for Sunflowers
  • Anup B. Rao
  • Computer Science, Mathematics
    discrete Analysis
  • 2020
TLDR
This work simplifies a recent result of Alweiss, Lovett, Wu and Zhang that gives an upper bound on the size of every family of sets of size $k$ that does not contain a sunflower and shows how to use the converse of Shannon's noiseless coding theorem to give a cleaner proof.
Improved lifting theorems via robust sunflowers
TLDR
This work develops a new approach to prove lifting theorems for the indexing gadget, based on a novel connection to the recently developed robust sunflower lemmas, and shows that this allows to reduce the gadget size to linear.
Monotone Circuit Lower Bounds from Robust Sunflowers
TLDR
A notion of robust clique-sunflowers is introduced and this is used to prove an $n^{\Omega(k)}$ lower bound on the monotone circuit size of the CLIQUE function for all $k \le n^{1/3-o(1)}$, strengthening the bound of Alon and Boppana.
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Coding for Sunflowers
  • Anup B. Rao
  • Computer Science, Mathematics
    discrete Analysis
  • 2020
TLDR
This work simplifies a recent result of Alweiss, Lovett, Wu and Zhang that gives an upper bound on the size of every family of sets of size $k$ that does not contain a sunflower and shows how to use the converse of Shannon's noiseless coding theorem to give a cleaner proof.
Monotone Circuit Lower Bounds from Robust Sunflowers
TLDR
A notion of robust clique-sunflowers is introduced and this is used to prove an $n^{\Omega(k)}$ lower bound on the monotone circuit size of the CLIQUE function for all $k \le n^{1/3-o(1)}$, strengthening the bound of Alon and Boppana.
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