Matroid coloring book

Color a Boolean lattice by a matroid rank function.

Ground set size n

Rank assignment

Choose a subset in the lattice to assign its rank value.

Consistency conditions enforced: 0 ≤ r(A) ≤ |A|, monotonicity, and submodularity.

3 rules of the coloring game

Choose a subset and assign its rank directly in the enlarged view.

Image: Anatoly Fomenko, A space with nontrivial local homology (1967), from Mathematical Impressions (AMS).