Linear Programming Relaxations of Maxcut

It is well-known that the integrality gap of the usual linear programming relaxation for Maxcut is 2 - epsilon. For general graphs, we prove that for any c and any fixed boundk, adding linear constraints of support bounded by k does not reduce the gap below 2-epsilon. We generalize this to prove that for any epsilon and any fixed bound k, strengthening the usual linear programming relaxation by doing k rounds of Sherali-Adams lift-and-project does not reduce the gap below 2 - epsilon. On the other hand, we prove that for dense graphs, this gap drops to 1 + epsilon after adding all linear constraints of support bounded by some constant depending on epsilon.