  Sharpening (a particular case of) a result of Szemeredi and Vu and extending
earlier results of Sarkozy and ourselves, we find, subject to some technical
restrictions, a sharp threshold for the number of integer sets needed for their
sumset to contain a block of consecutive integers of length, comparable with
the lengths of the set summands.
  A corollary of our main result is as follows. Let $k,l\ge 1$ and $n\ge 3$ be
integers, and suppose that $A_1,...,A_k\subset[0,l]$ are integer sets of size
at least $n$, none of which is contained in an arithmetic progression with
difference greater than 1. If $k\ge 2\lceil(l-1)/(n-2)\rceil$, then the sumset
$A_1+...+A_k$ contains a block of consecutive integers of length $k(n-1)$.
