Multiplicity of monochromatic solutions to

For integers n ⩾ 1 and k ⩾ 0 , let M k ( n ) represent the minimum number of monochromatic solutions to x + y < z over all 2-colorings of { k + 1 , k + 2 , … , k + n } . We show that for any k ⩾ 0 , M k ( n ) = C n 3 ( 1 + o k ( 1 ) ) , where C = 1 12 ( 1 + 2 2 ) 2 ≈ 0.005686 . A structural result is also proven, which can be used to determine the exact value of M k ( n ) for given k and n.