Let denote the set of degree monomials in variables, where in the multi-linear setting . The current works showed (implicitly) that there exist subsets such that any subset of size is contained in for some . This was done by choosing for all possible subsets of of size . This then implied a bound of on the rank of an appropriate matrix (or tensor, as given in the symmetric version in Terry’s blog).

However, the bound on can be improved to by a more careful choice. The idea is simple: let enumerate the intervals for , where values are taken modulo . For each such interval , take for all subsets of of size of it, and for all subsets of the complement of of size .

In general, it seems that this allows one to take $approx N \approx \sqrt{|M_{n,d}|}$ even in more general settings, which would improve the bounds for cap sets and the weak sunflower theorem.

]]>sunflowers show up in razborovs proof of the “monotone P vs NP” and in some other analysis eg Rossman. my question, could there be a conjecture on sunflowers or something similar that is equivalent to the P vs NP problem (or maybe P/poly vs NP)? ]]>

Thanks for pointing out the second comment by Eric Naslund. That is very nice. I think. If I read it correctly, this is a direct proof (at least a case of) of Conjecture 4 of Alon-Shpilka-Umans, which is equivalent to the Erdos and Szemeredi conjecture. So it is still a version of the the sunflower conjecture that allows the bound to grow with the size of some universe. In that sense I still have no idea how to obtain a polynomial bound that does not depend on the size of “some universe”.

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