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

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 32 768 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.