On the asymptotic minimum number of monochromatic 3-term arithmetic progressions

Let V(n)V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-coloring of {1,2,…,n}{1,2,…,n}. We show that167532768n2(1+o(1))⩽V(n)⩽1172192n2(1+o(1)). As a consequence, we find that V(n)V(n) is strictly greater than the corresponding number for Schur triples (which is 122n2(1+o(1))). Additionally, we disprove the conjecture that V(n)=116n2(1+o(1)) as well as a more general conjecture.