Hi BreakingBed,
This is only a general suggestion (less than 3") but smaller gaps would be better yet. The more the surface area of the support surface while still maintaining good airflow and the stronger the wood you use for the slats the better. The least possible amount of flex in other words is the goal.
I would tend to use 1x3’s (which are usually .75" x 2.5") or wider (you can cut your own from furring strip board). Again the more surface area to distribute the weight (while still maintaining good airflow) and the less flex in the support surface the better.
The foundation post here has several suggestions you can use that were shared by some of the members here (under the foundation and platform bed sections).
Phoenix