No cardinal function strictly smaller than ded bounds the traces of a family of finite VC dimension. For every infinite κ and every lam < ded κ there is a family of VC dimension 1 on a ground set of size exactly κ tracing more than lam sets. Together with HasVCDimLE.mk_image_inter_le_ded, which caps those traces at ded #A, this places ded at the exact strength of the bound.

The conclusion cannot be strengthened to a family tracing exactly ded κ sets. Chernikov and Shelah, On the number of Dedekind cuts and two-cardinal models of dependent theories, note after their definition of ded κ that "in general the supremum need not be attained", so at such a κ no family attains it and quantifying below the supremum is the available form. At κ = ℵ₀ the supremum is attained, by mk_image_inter_range_cut_eq_ded.