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Hyperplane arrangements, discrete geometry, convex polytopes, and optimization. For more general questions concerning matroids, use the matroid-theory tag.
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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 …