Long arithmetic progressions in sumsets: Thresholds and Bounds
  For a set $A$ of integers, the sumset $lA =A+...+A$ consists of those numbers which can be represented as a sum of $l$ elements of $A$   $$lA =\{a_1+... a_l| a_i \in A_i \}. $$   A closely related and equally interesting notion is that of $l^{\ast}A$, which is the collection of numbers which can be represented as a sum of $l$ different elements of $A$   $$l^{\ast} A =\{a_1+... a_l| a_i \in A_i, a_i \neq a_j \}. $$   The goal of this paper is to investigate the structure of $lA$ and $l^{\ast}A$, where $A$ is a subset of $\{1,2, ..., n\}$. As applications, we solve two conjectures by Erd\"os and Folkman, posed in sixties. 
E. Szemeredi and V. Vu