Traces and shattering
How many sets a family can cut out, and how many it shatters.
- MTH.C-2026-6001encard_image_inter_le_encard_shatters
Pajor's inequality, with no finiteness assumptions: the traces of
𝒜onAare at most as many as the subsets ofAshattered by𝒜. For a finite trace family this is a descent on the number of traces that never consumes the ground set; an infinite trace family forces infinitely many shattered singletons, and both sides are⊤. - MTH.C-2026-6002HasVCDimLE.vcGrowth_le_exp
- MTH.C-2026-6003mk_determined_le_mk_shatters
The cardinal Pajor inequality for determined traces: the traces of
𝒜onAthat are determined by finitely many points inject into the shattered subsets, with no hypotheses. Unlike theℕ∞-valuedencard_determined_le_encard_shatters, this bounds genuine cardinalities; the full trace family cannot replace the determined traces (not_mk_image_inter_le_mk_shatters). - MTH.C-2026-6004not_mk_image_inter_le_mk_shatters
The full trace family cannot replace the determined traces in
mk_determined_le_mk_shatters: the half-line cuts overℚtrace continuum-many sets while shattering only countably many.