  In previous work, we noted that the known cases of hyper-K\"ahler manifolds
satisfy a natural condition on the LLV decomposition of the cohomology;
informally, the Verbitsky component is the dominant representation in the LLV
decomposition. Assuming this condition holds for all hyper-K\"ahler manifolds,
we obtain an upper bound for the second Betti number in terms of the dimension.
