From coordinate subspaces over finite fields to ideal multipartite uniform clutters

Take a prime power q , an integer $$n\ge 2$$ n ≥ 2 , and a coordinate subspace $$S\subseteq GF(q)^n$$ S ⊆ G F ( q ) n over the Galois field GF ( q ). One can associate with S an n -partite n -uniform clutter $$\mathcal {C}$$ C , where every part has size q and there is a bijection between the vectors in S and the members of $$\mathcal {C}$$ C . In this paper, we determine when the clutter $$\mathcal {C}$$ C is ideal , a property developed in connection to Packing and Covering problems in the areas of Integer Programming and Combinatorial Optimization. Interestingly, the characterization differs depending on whether q is 2, 4, a higher power of 2, or otherwise. Each characterization uses crucially that idealness is a minor-closed property : first the list of excluded minors is identified, and only then is the global structure determined. A key insight is that idealness of $$\mathcal {C}$$ C depends solely on the underlying matroid of S . Our theorems also extend from idealness to the stronger max-flow min-cut property. As a consequence, we prove the Replication and $$\tau =2$$ τ = 2 Conjectures for this class of clutters.