Forbidden Intersections

About 10 years ago P. Erdos conjectured that if F is a family of subsets of {1,2..,n} without F.F' Epsilon F, /F F'/ =n/4 then /F/ (2-epsilon)n holds for some positive absolute constant epsilon Here this conjecture is proved in a stronger form (Theorem 1.1), which solves a $250 problem of Erdos [E2]. Suppose C is a code (i.e., a collection of codewords) of length n over an alphabet of q elements, 1/2 > Delta > 0 is arbitrary. Suppose further that there are no two codewords at Hamming distance d where d is a fixed integer, Delta n (1-Delta) n and d is even if q=2. Then /C/ (q-epsilon)n where epsilon > 0 depends on q and Delta.