In In mattress covers the cloth is referred to as ticking.
In sofa bed mattresses, there is a design issue.
We would like to use a good quilted ticking. Unfortunately, when the mattress is folded away inside the
sofa, the quilted material tends to bunch up in the creases. The bunching causes other problems too. The
mattress cover and shift around so that the seams wind up moving onto the flat sleeping surface.
Experience has taught us that a flat, non-quilted ticking gives the most satisfactory long term service.