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Updated Oct 8

Problem 20

Statement. 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 $c_k>0$ ?

Status. Open.

Source. erdosproblems.com/20, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #20, https://www.erdosproblems.com/20.

References.

[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).

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (32)

Set Systems, Designs and Hypergraphs set_systems/alon_2013_sunflowers_matrix_multiplication Theorem 2.3 (p. 3): the uniform 3-sunflower bound c^s implies the {0,1}^n conjecture Theorem 2.6 (p. 5): the classical sunflower conjecture is equivalent to the one in Z_D^n set_systems/alweiss_2020_improved_bounds_sunflower_lemma Lemma 3.1: a w-set system of size ((log w)/8)^{w-sqrt w} with no (1/2,1/2)-robust sunflower Theorem 1.4: a w-set system of size (C r^3 log w log log w)^w contains an r-sunflower Theorem 1.9: a large w-uniform system contains an (alpha,beta)-robust sunflower Theorem 2.5: the spreadness threshold kappa(w,alpha,beta) is O(alpha^{-2}(log w log log w + (log 1/beta)^2))set_systems/bell_2021_note_sunflowers Lemma 2 (p. 1): a (Cp log k)-spread family of at least (Cp log k)^k k-sets has p disjoint members Lemma 4 (p. 2): the spread hypothesis of Theorem 3 is essentially optimal Theorem 1 (p. 1): Sun(p,k) is at most (Cp log k)^k Theorem 3 (p. 2): the spread estimate of Rao and Tao set_systems/erdos_1960_intersection_theorems_systems_sets Conjecture (p. 86): b! in the sunflower bound replaced by c_1^b Theorem I (p. 86): the Δ-system lemma for arbitrary cardinals Theorem II (p. 86): an (a^{b+1}, b)-system with no Δ(>a)-system Theorem III (p. 86): the finite sunflower lemma set_systems/erdos_1981_combinatorial_problems_which_i_would_most set_systems/frankston_2019_thresholds_versus_fractional_expectation_thresholds Lemma 3.1 (p. 5): a random np-set W leaves few bad pairs (S, W), on average at most |H| C^{-r/3}Theorem 1.1 (p. 1): p_c(F) <= K q_f(F) log l(F)set_systems/kostochka_1999_systems_small_sets_no_large_subsystems Theorem 1 (p. 266): for fixed r, f(r,k) = k^r + o(k^r) as k grows Theorem 2 (p. 266): for fixed r and large k, f(r,k) <= k^r(1 + c_r k^{-2^{-r}})set_systems/naslund_2017_upper_bounds_sunflower_free_sets Theorem 5 (p. 2): a sunflower-free set in (Z/DZ)^n has at most c_D^n elements, c_D = 3(D-1)^{2/3}/2^{2/3}set_systems/rao_2020_coding_sunflowers Lemma 2: an r(p,k)-spread sequence of more than r(p,k)^k sets of size k contains p disjoint sets Lemma 4: a random set of size gamma n likely contains a set of a long r-spread sequence Theorem 1: more than (alpha p log(pk))^k sets of size k contain a p-sunflower

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