Short rainbow cycles in graphs and matroids

Let (Formula presented.) be a simple (Formula presented.) -vertex graph and (Formula presented.) be a coloring of (Formula presented.) with (Formula presented.) colors, where each color class has size at least 2. We prove that (Formula presented.) contains a rainbow cycle of length at most (Formula presented.), which is best possible. Our result settles a special case of a strengthening of the Caccetta-Häggkvist conjecture, due to Aharoni. We also show that the matroid generalization of our main result also holds for cographic matroids, but fails for binary matroids.