Cotton bed sheet is fine. I used that before I found Empress mills packs of white cotton. (Actually only painted once but it's on my list of things to get to grips with)
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You need a piece of board covered in plastic to lay your fabric on. A spray bottle for water to dampen fabric and hot sun to dry it........ha ha You can iron the painted fabric or put it in the oven to set it. I don't think you need the chemicals you use with dying because the paint sits on the fabric rather than being drawn into it.
I'll see if I can find the links to videos that go through it. Aargh you are going to get me painting when I should be finishing projects
Mickey Lawler showed painting fabrics in a recent show but can't remember if she showed the whole process.