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As you may know, there has been very recently a big breakthrough concerning upper bounds for the capset problem over $\mathbb{F}_3^n$ (and further generalizations to $\mathbb{F}_q^n$). I was wondering which other configurations have been studied so far in this context. For instance, the corner problem has also been studied, but it does not arise from a linear equation.

Also, on the other direction, lower bounds for the capset problem come from a work by Edel, so I would like to know what is known for other configurations in the finite field setting.

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    $\begingroup$ This is a rather broad question, a complete answer to which would constitute a good survey paper. My best advice would be to check the recent papers of Peter Pach and Cristian Elsholtz (they even have a common paper "Caps and progression-free sets in Znm", in Des. Codes Cryptogr. 88 (2020), no. 10, 2133–2170). Yet another paper to look at: F. Petrov and C. Pohoata, "Improved bounds for progression-free sets in Cn8", Israel J. Math. 236 (2020), no. 1, 345–363. $\endgroup$
    – Seva
    Apr 25 at 13:05

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The methods of Edel's lower bound, which I have improved in this paper, are not specific to the setting of $\mathbb{F}_3^n$ as far as I can see. My result and Edel's both come from considering 3 cap sets $A_0, A_1, A_2 \subseteq \mathbb{F}_3^6$ and then extending them, but these ideas can be used for any $\mathbb{F}_p^n$, I believe. Taking direct products and their unions, which is the construction of Edel, should be possible in general characteristic.

It does depend what you mean by the problem in other finite fields, since cap set is a definition specific to $\mathbb{F}_3^n$. Do you mean no non trivial solutions to $x+y+z=0$, no non trivial solutions to $x+y=2z$, no 3APs or no lines? In characteristic 3, these are equivalent, but they are not in general!

The no lines definition is often called a 'cap' in the design theory and finite geometry literature, and is the object of study in Edel's paper. He does state and prove everything for general finite fields, but the only examples of affine caps he discusses are in characteristic 3. By contrast, my paper only studies the setting of $\mathbb{F}_3^n$.

The construction relies on 2 things:

  1. A collection of cap sets, which satisfy certain structural conditions (I call this 'extendable', Edel calls it 'property $E_L$')
  2. Indexing sets (which I call 'admissible sets', but Edel calls 'capsets')

Depending on what you mean by 'other configurations in the finite field setting', I think a version of the extended product of Edel should be possible, but it requires good examples from low dimensions, which are mostly (I think) only known for cap sets (that is, in $\mathbb{F}_3^n$).

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