Before revision:  On the positive side,  we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf  {\sf  {\sf   .
After revision:  On the positive side, we provide subquadratic kernels on H-free graphs for several graphs H, such as the bull, the paw, or the complete graphs, by making use of the Erdos-Hajnal property in order to find an appropriate decomposition {\sf {\sf {\sf poly .