# 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…

## 54 Citations

Turán numbers of sunflowers

- Mathematics
- 2021

A collection of distinct sets is called a sunflower if the intersection of any pair of sets equals the common intersection of all the sets. Sunflowers are fundamental objects in extremal set theory…

Tur\'{a}n numbers of sunflowers

- Mathematics
- 2021

A collection of distinct sets is called a sunflower if the intersection of any pair of sets equals the common intersection of all the sets. Sunflowers are fundamental objects in extremal set theory…

Sunflowers in Set Systems of Bounded Dimension

- MathematicsSoCG
- 2021

The Erdős-Rado conjecture for families F of bounded Littlestone dimension and for some geometrically defined set systems is verified.

Lifting with Sunflowers

- Computer Science, MathematicsITCS
- 2022

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.

Near-sunflowers and focal families

- Mathematics
- 2020

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…

Sunflowers and Robust Sunflowers from Randomness Extractors

- Mathematics
- 2022

The Erdős–Rado Sunflower Theorem (J. London Math. Soc. 1960) is a fundamental result in combinatorics, and the corresponding Sunflower Conjecture is a central open problem. Motivated by applications…

On $t$-intersecting Hypergraphs with Minimum Positive Codegrees

- Mathematics
- 2021

For a hypergraph H, define the minimum positive codegree δ i (H) to be the largest integer k such that every i-set which is contained in at least one edge of H is contained in at least k edges. For 1…

Coding for Sunflowers

- Computer Science, Mathematicsdiscrete Analysis
- 2020

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

- Computer Science, MathematicsElectron. Colloquium Comput. Complex.
- 2020

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

- Mathematics, Computer ScienceLATIN
- 2020

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

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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.

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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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