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

Abstract

Let V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-coloring of {1,2,...,n}. We show that (1675/32768) n^2 (1+o(1)) <= V(n) <= (117/2192) n^2(1+o(1)). As a consequence, we find that V(n) is strictly greater than the corresponding number for Schur triples (which is (1/22) n^2 (1+o(1)). Additionally, we disprove the conjecture that V(n) = (1/16) n^2(1+o(1)), as well as a more general conjecture.