Packing and covering in infinite matroids

OpenAI's theorem (2026) that the infinite matroid packing/covering conjecture is false in ZFC; through density ideals, an ultrafilter with an ordinal rank, local bases defined by probes, and a self-dual uniform matroid placed along a double ray.

Five lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

  1. Infinite matroids and the packing/covering conjecture
  2. Ideals, ultrafilters and an ordinal rank
  3. Probes and local bases
  4. Assembling a self-dual matroid
  5. No independent covering

Theorems and assumptions · Reader and editable sources · Course record

The first lesson proves general duality, restriction and contraction for infinite matroids and the intersection equivalence; the universal packing and covering equivalences are proved with the wave and family-encoding arguments.

Prerequisites