In the framework of agnostic learning, one of the main open problems of the theory of multi-category pattern classification is the characterization of the way the complexity varies with the number C of categories. More precisely, if the classifier is characterized only through minimal learnability hypotheses, then the optimal dependency on C that an upper bound on the probability of error should exhibit is unknown. We consider margin classifiers. They are based on classes of vector-valued functions with one component function per category, and the classes of component functions are uniform Glivenko-Cantelli classes. For these classifiers, an L p-norm Sauer-Shelah lemma is established. It is then used to derive guaranteed risks in the L $\infty$ and L 2-norms. These bounds improve over the state-of-the-art ones with respect to their dependency on C, which is sublinear.