Hi blackviolet,
A foam mattress with a polyfoam support core will generally do best with a firm, flat, and evenly supportive support surface underneath it that has minimal to no flex under the mattress and for larger sizes with at least one center support beam that has good support to the floor to prevent any sagging in the middle of the mattress. The components (either a bedframe and foundation or a platform bed) need to be strong and durable enough to support the weight of the mattress and the people sleeping on it without some of the parts bending, sagging, shifting, or breaking with extended use. The support surface under the mattress (which may be slats or a steel or wire grid) should have enough surface area to prevent the mattress from sagging through any gaps or spaces in the support surface over time but still allow some airflow under the mattress. If a foundation has a slatted surface then the gaps between any slats should be no more than about 5" (with 1 x 3 slats) although I would suggest 4" or less which would be better yet.
There is also more information and some guidelines about the different type of support systems (bedframes and foundations or platform beds) that are generally suitable for different types of mattresses and some examples of each of them in post #1 here (the first post in the topic we are posting in) that may be helpful.
If you are looking for a specific height for your sleeping surface then you would need to add the height of your bedframe and foundation (or the height of a platform bed that doesn’t need a foundation) to the thickness of your mattress and each of them are available in different heights.
Platform beds and bed frames are more of a furniture item and aren’t something that I spend a great deal of time researching in detail since there are so many different options available, personal preferences about the look of furniture play such a significant role in different people’s choices … and research about mattresses take up most of my available time.
Having said that … some of the suggestions in the reference post I linked may meet your criteria but I don’t know the heights of each of them off the top of my head so you would need to check their websites or make a few phone calls to see if the components in your support system in combination with the thickness of your mattress add up to the height you are looking for.
Phoenix