Solution
= Solution
No. Take
$$
(a_1,a_2,a_3,a_4)=(1,1,-1,-1).
$$
The full set of coefficients sums to zero, so the equation is partition regular by the <Rado theorem for one equation>. Every solution satisfies
$$
x_1+x_2-x_3-x_4=0,
$$
which can never be positive.