I do not know of a direct connection to Roth or Szemerédi over the integers.  However, the paper

>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/~nogaa/PDFS/sfmmccc2.pdf))

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 


> 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)).