Ellenberg and Gijswijt showed that the largest subset of $mathbb{F}_q^n$ with no three terms in arithmetic progression has size $c^n$ where $c<q$.

Ellenberg and Gijswijt actually proved a generalisation:

Let $x_1, x_2, x_3in S subset mathbb{F}_q^n$ then

$a_1x_1+a_2x_2+a_3x_3=0$ with $a_1+a_2+a_3=0$ has no solutions (excluding the trivial $x_1=x_2=x_3$) $implies$ $|S|<c^n$ where $c<q$.

(Choosing $(a_1,a_2,a_3)=(1,-2,1)$ gives the case of AP’s)

My question is:

**How do the above bounds for |S| improve if we add additional equations and ask that all of them individually have no solutions?**

Specifically consider a set of equations $$a_{i1}x_1+a_{i2}x_2+a_{i3}x_3=0,$$ $1 leq i leq m$ with $a_{i1}+a_{i2}+a_{i3}=0$, distinct (up to multiplication by a constant) and the $a_{ij}$‘s all non-zero.

If $x_iin S subset mathbb{F}_q^n$. How large can $|S|$ be if none of the above equations have a (non-trivial) solution?

**Note that my condition is not that the system of equations as a whole has no solution but rather the much stronger assertion that none of the equations of the system individually can be satisfied non-trivially.**

Any guesses, heuristics, plausible conjectures are welcome.