Let p be a prime, n a natural number, and S a set of cardinality pn . Let <spanclass=′latex−bold′>P</span> be a family of partitions of S into nonempty parts of sizes divisible by p such that the intersection of any two parts that occur in any of the partitions has at most one element. How large can ∣<spanclass=′latex−bold′>P</span>∣ be? advanced fieldsadvanced fields unsolved