Long arithmetic progressions in sumsets: Thresholds and Bounds

For a set $A$ of integers, the sumset $lA=A+\mathrm{.}\mathrm{.}\mathrm{.}+A$ consists of those numbers which can be represented as a sum of $l$ elements of $A$ $$lA=\{a_1+\mathrm{.}\mathrm{.}\mathrm{.}{a}_l\mathrm{\mid }{a}_i\in A_i\}\mathrm{.}$$ 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 $A$ $$l^{\ast }A=\{a_1+\mathrm{.}\mathrm{.}\mathrm{.}{a}_l\mathrm{\mid }{a}_i\in A_i,a_i\mathrm{\ne }{a}_j\}\mathrm{.}$$ The goal of this paper is to investigate the structure of $lA$ and $l^{\ast }A$, where $A$ is a subset of $\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},n\}$. As applications, we solve two conjectures by Erd"os and Folkman, posed in sixties.