Let $S$ be a family of sets.
We say a subset $S'\subseteq S$ is good if we can choose from every set $A\in S'$ a representative $x_A$ s.t.: For every three sets $A,B,C\in S'$ it holds that $(x_A + x_B + x_C)$ mod $10 \neq 0$.
Using the compactness theorem I've shown that if all sets $A\in S$ are finite, and if every finite subset $S'\subseteq S$ is good then $S$ is also good.
I've used the finite property of the elements in $S$ in my proof, but I can't think of an example where letting $S$ have infinite sets leads to a contradiction.
Any thoughts?
I just reduced the problem of infinite sets to finite sets (all the set has to contain is the remainders from division by 10).