Matroids
What is a Matroid?
A matroid is a pair $(S,\mathcal{I})$ where $S$ is a set and $\mathcal{I}$ is a set of subsets of $S$ where (a) $\varnothing \in \mathcal{I}$, (b) if $A \subseteq B \in \mathcal{I}$ then $A \in \mathcal{I}$, and (c) if $A,B \in \mathcal{I}$ and $|A| < |B|$, then there is an $x \in B - A$ such that $A \cup\{x\} \in \mathcal{I}$.
The sets in $\mathcal{I}$ are called independent. Refer to standard references for a more extensive introduction.