I do not know of a direct connection to Roth or Szemerédi over the integers.  However, https://www.tau.ac.il/~nogaa/PDFS/sfmmccc2.pdf shows that a proof of the Sunflower Conjecture would imply 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 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 is equivalent to 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 https://core.ac.uk/download/pdf/82090028.pdf).