Hi reijam,
Congratulations on your new mattress … you certainly made a great quality/value choice 
The 3" guidelines applies more to mattresses that have a latex support core and mattresses with a polyfoam support core are stiffer and will be OK with gaps that are a little wider (although smaller gaps are still better and more evenly supportive). There is more about the types of support systems and foundations that are best for different types of mattresses in the foundation post here and the posts it links to (in the second paragraph).
In general the “bottom line” goal for a support system for a foam mattress is to have the least possible flex, good airflow under the mattress, good center support, and an evenly supportive surface that doesn’t allow the mattress to sag into any gaps. Part of this would depend on the number, type, thickness, and width of the slats but part of this also depends on the quality and relative strength of the wood. There is more information about the relative bending strength and stiffness of different types of wood here that should be helpful.
Phoenix