A family of subsets of an n-set is k-locally thin if, for every k-tuple of its members, the ground set has at least one element contained in exactly one of them. For k = 5 we derive a new exponential upper bound for the maximum size of these families. This implies the same bound for all odd values of k > 3. Our proof uses the graph entropy bounding technique to exploit a self-similarity in the structure of the hypergraph associated with such set families.