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Questions related to the field of Combinatorics called Matroid Theory. Relevant topics include matroids in Combinatorial Optimization, Lattice Theory, Algebraic Geometry, Polyhedral Theory, Rigidity, and Algorithms. For questions about Oriented Matroids, the oriented-matroids tag may be used.
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Small remarkable matroids
Other than those you've listed, the most obvious "interesting" matroids I can think of on up to 7 elements are:
$M(K_4)$, the cycle matroid of $K_4$,
$W^3$, the "whirl" obtained from $M(K_4)$ by rel …