Problem 20

Let $f(n,k)$ be minimal such that every family $\mathcal{F}$ of $n$ -uniform sets with $\lvert \mathcal{F}\rvert \geq f(n,k)$ contains a $k$ -sunflower. Is it true that

f(n,k) < c_k^n

for some constant $c_k>0$ ?

Source: erdosproblems.com/20

Combinatorics

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Open

No claim settles this problem.

ALWZ20: Alweiss, R. and Lovett, S. and Wu, K. and Zhang, J., Improved bounds for the sunflower lemma. (2020).
BCW21: Bell, T. and Chueluecha, S. and Warnke, L., Note on sunflowers. Discret. Math. (2021).
Er81: Erdős, P., On the combinatorial problems which I would most like to see solved. Combinatorica (1981), 25-42.
ErRa60: Erdős, P. and Rado, R., Intersection theorems for systems of sets. J. London Math. Soc. (1960), 85-90.
FKNP19: Frankston, K. and Kahn, J. and Narayanan, B. and Park, J., Thresholds versus fractional expectation-thresholds. CoRR (2019).
KRT99: Kostochka, A. V. and Rödl, V. and Talysheva, L. A., On systems of small sets with no large $\Delta$ -subsystems. Combin. Probab. Comput. (1999), 265-268.
Ko97: Kostochka, A., A bound on the cardinality of families not containing $\Delta$ -systems. (1997).
Ra20: Rao, A., Coding for sunflowers. Discrete Analysis (2020).

Problem 20

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