Return to Answer

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 (author 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 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

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 (Core pdf).

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 (author 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 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 (Core pdf).

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 (author 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 this 2016 blog post by Gil Kalai 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

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 (Core pdf).

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 (author 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 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

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 (Core pdf).

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).

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 (author 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 this 2016 blog post by Gil Kalai 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

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 (Core pdf).