RECORD
Asymptotic Improvement of the Sunflower Bound
Combinatorics · Submitted 15 Aug 2014
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim This paper shows that the general sunflower bound can be indeed reduced by an exponential factor: We prove that ${\cal F}$ includes a sunflower of cardinality $k$ if \[ |{\cal F|} \ge \left( \sqrt{10} -2 \right)^2 \left[ k \cdot \min \left( \frac{1}{\sqrt{10}-2}, \frac{c}{\log \min(k, s)} \right) \right]^s s!, \] for a constant $c>0$, and any $k \ge 2$ and $s \ge 2$.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
A sunflower with a core $Y$ is a family ${\cal B}$ of sets such that $U \cap U' = Y$ for each two different elements $U$ and $U'$ in ${\cal B}$. The well-known sunflower lemma states that a given family ${\cal F}$ of sets, each of cardinality at most $s$, includes a sunflower of cardinality $k$ if $|{\cal F}|> (k-1)^s s!$. Since Erd\"os and Rado proved it in 1960, it has not been known for more than half a century whether the sunflower bound $(k-1)^s s!$ can be improved asymptotically for any $k$ and $s$. It is conjectured that it can be reduced to $c_k^s$ for some real number $c_k>0$ depending only on $k$, which is called the sunflower conjecture. This paper shows that the general sunflower bound can be indeed reduced by an exponential factor: We prove that ${\cal F}$ includes a sunflower of cardinality $k$ if \[ |{\cal F|} \ge \left( \sqrt{10} -2 \right)^2 \left[ k \cdot \min \left( \frac{1}{\sqrt{10}-2}, \frac{c}{\log \min(k, s)} \right) \right]^s s!, \] for a constant $c>0$, and any $k \ge 2$ and $s \ge 2$. For instance, whenever $k \ge s^\epsilon$ for a given constant $\epsilon \in (0,1)$, the sunflower bound is reduced from $(k-1)^s s!$ to $(k-1)^s s! \cdot \left[ O \left( \frac{1}{\log s} \right) \right]^s$, achieving the reduction ratio of $\left[ O \left( \frac{1}{\log s} \right) \right]^s$. Also any ${\cal F}$ of cardinality at least $\left(\sqrt{10}-2 \right)^2 \left( \frac{k}{\sqrt{10}-2} \right)^s s!$ includes a sunflower of cardinality $k$, where $\frac{1}{\sqrt{10}-2}=0.8603796\ldots$. Our result demonstrates that the sunflower bound can be improved by a factor of less than a small constant to the power $s$, giving hope for further update.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | A sunflower with a core $Y$ is a family ${\cal B}$ of sets such that $U \cap U' = Y$ for each two different elements $U$ and $U'$ in ${\cal B}$. | background 1.00 | 0.05 | 0.01 | asserted 0.97 |
| s1 | The well-known sunflower lemma states that a given family ${\cal F}$ of sets, each of cardinality at most $s$, includes a sunflower of cardinality $k$ if $|{\cal F}|> (k-1)^s s!$. | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s2 | Since Erd\"os and Rado proved it in 1960, it has not been known for more than half a century whether the sunflower bound $(k-1)^s s!$ can be improved asymptotically for any $k$ and $s$. | gap 0.92 | 0.04 | 0.02 | asserted 0.94 |
| s3 | It is conjectured that it can be reduced to $c_k^s$ for some real number $c_k>0$ depending only on $k$, which is called the sunflower conjecture. | background 0.90 | 0.03 | 0.03 | speculative 0.98 |
| s4 | This paper shows that the general sunflower bound can be indeed reduced by an exponential factor: We prove that ${\cal F}$ includes a sunflower of cardinality $k$ if \[ |{\cal F|} \ge \left( \sqrt{10} -2 \right)^2 \left[ k \cdot \min \left( \frac{1}{\sqrt{10}-2}, \frac{c}{\log \min(k, s)} \right) \right]^s s!, \] for a constant $c>0$, and any $k \ge 2$ and $s \ge 2$. | result 0.88 | 0.97 | 0.93 | asserted 1.00 |
| s5 | For instance, whenever $k \ge s^\epsilon$ for a given constant $\epsilon \in (0,1)$, the sunflower bound is reduced from $(k-1)^s s!$ to $(k-1)^s s! | result 0.93 | 0.84 | 0.11 | asserted 0.93 |
| s6 | \cdot \left[ O \left( \frac{1}{\log s} \right) \right]^s$, achieving the reduction ratio of $\left[ O \left( \frac{1}{\log s} \right) \right]^s$. | result 0.91 | 0.90 | 0.13 | asserted 0.99 |
| s7 | Also any ${\cal F}$ of cardinality at least $\left(\sqrt{10}-2 \right)^2 \left( \frac{k}{\sqrt{10}-2} \right)^s s!$ includes a sunflower of cardinality $k$, where $\frac{1}{\sqrt{10}-2}=0.8603796\ldots$. | result 0.99 | 0.92 | 0.24 | asserted 1.00 |
| s8 | Our result demonstrates that the sunflower bound can be improved by a factor of less than a small constant to the power $s$, giving hope for further update. | interpretation 0.96 | 0.89 | 0.28 | asserted 0.98 |
What was examined, and what was not
- arXiv listing abstract v4
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
A sunflower with a core $Y$ is a family ${\cal B}$ of sets such that $U \cap U' = Y$ for each two different elements $U$ and $U'$ in ${\cal B}$. The well-known sunflower lemma states that a given family ${\cal F}$ of sets, each of cardinality at most $s$, includes a sunflower of cardinality $k$ if $|{\cal F}|> (k-1)^s s!$. Since Erd\"os and Rado proved it in 1960, it has not been known for more than half a century whether the sunflower bound $(k-1)^s s!$ can be improved asymptotically for any $k$ and $s$. It is conjectured that it can be reduced to $c_k^s$ for some real number $c_k>0$ depending only on $k$, which is called the sunflower conjecture. This paper shows that the general sunflower bound can be indeed reduced by an exponential factor: We prove that ${\cal F}$ includes a sunflower of cardinality $k$ if \[ |{\cal F|} \ge \left( \sqrt{10} -2 \right)^2 \left[ k \cdot \min \left( \frac{1}{\sqrt{10}-2}, \frac{c}{\log \min(k, s)} \right) \right]^s s!, \] for a constant $c>0$, and any $k \ge 2$ and $s \ge 2$. For instance, whenever $k \ge s^\epsilon$ for a given constant $\epsilon \in (0,1)$, the sunflower bound is reduced from $(k-1)^s s!$ to $(k-1)^s s! \cdot \left[ O \left( \frac{1}{\log s} \right) \right]^s$, achieving the reduction ratio of $\left[ O \left( \frac{1}{\log s} \right) \right]^s$. Also any ${\cal F}$ of cardinality at least $\left(\sqrt{10}-2 \right)^2 \left( \frac{k}{\sqrt{10}-2} \right)^s s!$ includes a sunflower of cardinality $k$, where $\frac{1}{\sqrt{10}-2}=0.8603796\ldots$. Our result demonstrates that the sunflower bound can be improved by a factor of less than a small constant to the power $s$, giving hope for further update.
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.