Let $n\geq 2$ be a natural number and consider the following: 


**$AC(n)$:** For each family $\{X\_i\}\_{i \in I}$ of $n$-element sets the product $\prod\_{i\in I}X\_i$ is non-empty. 


Is it known that for which values of $m$ and $n$, **$AC(m)$** and **$AC(n)$** are equivalent !? 



Weak forms of the Axiom of Choice
There is about half a chapter devoted to this in Jech **The Axiom of Choice**, the key point is Theorem 7.15 which gives a condition for $\mathsf{AC}(n)\implies\mathsf{AC}(m)$. You may want to look around there (p.111 for the theorem).


