I have observed some similar things between a reformulation of the [Sunflower conjecture](https://en.wikipedia.org/wiki/Sunflower_(mathematics)#Sunflower_lemma_and_conjecture) (see also conjecture 1.3 in *[Improved bounds for the sunflower lemma](https://arxiv.org/abs/1908.08483)*) and [Szemerédi's theorem](https://en.wikipedia.org/wiki/Szemer%C3%A9di%27s_theorem) such that for Szemerédi's theorem we have an often-used equivalent finitary version which states that:



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> **Theorem**: for every positive integer $k$ and real number $\delta\in (0,1]$ there exists a positive integer $N=N(\delta,k)$ such that every subset of $\{1,2,\cdots,N\}$ of size at least $δN$ contains an arithmetic progression of length $k$ say $k\text{-arithmetic progression}$
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The Sunflower conjecture states that :



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> **Conjecture**: Let $r ≥ 3$. There exists $c = c(r)$ such that any $w$-set system $F$ of size $|F| ≥ c^w$ contains an $r$-sunflower
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We see that both of them investigate about existence of such constant with such size bounds for which we have $k\text{-arithmetic progression}$ or $r$-sunflower. Also, another thing that attracted my attention is that for [Erdős conjecture on arithmetic progressions](https://en.wikipedia.org/wiki/Erd%C5%91s_conjecture_on_arithmetic_progressions), Erdős and Turán made in 1936 the weaker conjecture that any set of integers with positive natural density contains infinitely many 3-term arithmetic progressions. This was proven by Klaus Roth in 1952, and generalized to arbitrarily long arithmetic progressions by Szemerédi in 1975 in what is now known as Szemerédi's theorem. In the opposite direction, **Kostochka** proved that any $w$-set system of size
$$
|F| \geq 
 cw! \cdot {(\log \log \log w/ \log \log w)}^{w}
$$
must contain a $3$-sunflower for some absolute constant $c$. Now for quantitative bounds of $r\_k(N)$(the size of the largest subset of $\{1, 2, \ldots, N \}$ without an arithmetic progression of length $k$) it is an open problem to determine its exact growth rate. In the same time it is an open problem to determine the exact rate growth or exact bounds for [The sunflower](https://terrytao.wordpress.com/2020/07/20/the-sunflower-lemma-via-shannon-entropy/#more-11852) (the sunflower lemma via Shannon entropy). My question here is:



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> **Question**: According to similar things which I have cited above between the Sunflower conjecture and Szemerédi's theorem, is there any non-trivial relationship between them? And in which context we can consider Sunflower coincide with arithmetic progression?
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Is there any relationship between Szemerédi's theorem and Sunflower conjecture?
I do not know of a direct connection to Roth or Szemerédi over the integers. However, the paper



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> N. Alon, A. Shpilka and C. Umans, *On Sunflowers and Matrix Multiplication*, 2012 IEEE 27th Conference on Computational Complexity, Porto, 2012, pp. 214-223, doi:[10.1109/CCC.2012.26](https://doi.org/10.1109/CCC.2012.26) ([author pdf](https://www.tau.ac.il/%7Enogaa/PDFS/sfmmccc2.pdf))
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shows that a proof of the Sunflower Conjecture would imply a proof of the Erdos-Szemeredi sunflower theorem, which also follows from a bound of $(3-\delta)^n$ for the capset problem, a strong form of Roth over $\mathbb{F}\_3^n$ (which is already known due to Croot-Lev-Pach). See [this 2016 blog post by Gil Kalai](https://gilkalai.wordpress.com/2016/05/17/polymath-10-emergency-post-5-the-erdos-szemeredi-sunflower-conjecture-is-now-proven/) for more discussions along this line.


It is also noteworthy that the Erdős-Szemerédi sunflower conjecture (which has been proved and follows from the capset problem) also implies that if $|S|=C\log(n)$ is a subset of $[n]$ for a large constant $C$, then there are three disjoints $X, Y, Z$ whose subset sums are identical, and thus the sums of the subsets $X, X \cup Y, X \cup Y \cup Z$ are in arithmetic progression; see



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> P. Erdős, A. Sárközy, *Arithmetic progressions in subset sums*, Discrete Mathematics **102** Issue 3 (1992) pp 249–264, doi:[10.1016/0012-365X(92)90119-Z](https://doi.org/10.1016/0012-365X%2892%2990119-Z)
> ([Core pdf](https://core.ac.uk/download/pdf/82090028.pdf)).
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