Complete Sequences inN2

Folkmanʹs theorem states that ifA={a1<a2<· · ·}satisfyingA(n)>n1/2+ϵ, whereA(n)=∑ai≤n1), thenP(A)={ai1+ai2+ · · ·+ais: i1<i2< · · ·<is;s∈N;aij∈A}contains an infinite arithmetic progression. We shall consider the analogue of Folkmanʹs theorem inN2, and some related questions are also investigated.