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 𝒜 on A are at most as many as the subsets of A shattered by 𝒜. 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 𝒜 on A that are determined by finitely many points inject into the shattered subsets, with no hypotheses. Unlike the ℕ∞-valued encard_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 over ℚ trace continuum-many sets while shattering only countably many.