Multiclass dimensions

Pajor's inequality beyond two labels, and where Natarajan and DS dimension part ways.

• MTH.C-2026-6005encard_image_restrict_le_encard_pairCubes

  Pajor's inequality for multiclass concept classes, with no hypotheses on the domain, the label type or the family: the traces of 𝒞 on S are at most as many as the pair-cubes of 𝒞 supported inside S.

• MTH.C-2026-6006exists_hasNatarajanDimLE_not_hasDSDimLE

  HasDSDimLE.hasNatarajanDimLE has no converse. The six-cycle on a two-point domain has Natarajan dimension 1, a Natarajan witness on both points being a four-cycle, while its whole trace family is a two-dimensional pseudo-cube, so its DS dimension is at least 2.