A survey on flows in graphs and matroids

A quintessential minimax relation is the Max-Flow Min-Cut Theorem which states that the largest amount of flow that can be sent between a pair of vertices in a graph is equal to the capacity of the smallest bottleneck separating these vertices. We survey generalizations of these results to multi-commodity flows and to flows in binary matroids. Two tantalizing conjectures by Seymour on the existence of fractional and integer flows are motivating our work. We will not assume from the reader any background in matroid theory.