Proving there exist three different vectors that sum to zero

Question

I'm trying to prove the following statement:

Prove that for each $S \subset {\mathbb{Z}_3}^3$ satisfaying $|S|=9$, there exist three different vectors $a, b, c \in S$ such that $a+b+c=\overrightarrow{0}$.

I know that a direct verification (which may be done by a computer) is an "immediate" proof. However, I'm interested in a simpler and more sophisticated proof. I've tried several approaches;

The first one is linear algebra - I've placed the elements of $S$ in a $9 \times 3$ matrix and tried to show that there exist three rows which sum to zero using the fact that the rank of this matrix $\leq 3$. It didn't work. The second approach I've tried is the pigeonhole principle, and it also didn't work. The third approach I've tried is the probabilistic method by randomly selecting three different vectors $a,b,c \in S$ and examining the three-dimensional random variable $a+b+c$, using the fact that it equals the zero vector iff each coordinate is zero.

I've also tried considering two cases: $\overrightarrow{0} \in S$ and $\overrightarrow{0} \not \in S$.

I've proved that there exists some coordinate $1 \leq i \leq 3$ such that there exist three vectors $a,b,c \in S$ such that their $i$th coordinates are $-1, 0, 1$. It might be useful. Thanks in advance.

Answer 1

$a+b+c=\vec{0}$ holds in $\Bbb Z_3^3$ if and only if for every component $i=0,1,2$, $$ a_i, b_i, c_i$$ are all equal or all distinct, and that is equivalent to $(a, b, c)$ being collinear in $\Bbb Z_3^3$ (as an affine space).

A Cap set is a set in $\Bbb Z_3^n$ with no three elements in a line, and the “cap set problem” is the problem of finding the size of the largest possible cap set, as a function of $n$.

Using that terminology, your claim is that

A subset $S$ of $\Bbb Z_3^3$ with 9 elements is not a cap set.

and that is wrong. A counterexample is: $$ (0, 0, 0), (0,0, 2), (0,2, 0),(0, 2, 2) ,\\ (1, 1, 1),\\ (2, 0, 1), (2, 1, 0), (2, 1,2), (2, 2, 1) \, . $$

This is the “3-D maximal cap” from Figure 11 in The Joy of SET:

But $9$ is the largest possible size of a cap set in $\Bbb Z_3^3$, compare Wikipedia: Cap set and A090245, which means that your statement becomes true if the condition $|S| = 9$ is replaced by $|S| = 10$.