  We prove a known 2-coloring of the integers $[N] := \{1,2,3,\ldots,N\}$
minimizes the number of monochromatic arithmetic 3-progressions under certain
restrictions. A monochromatic arithmetic progression is a set of equally-spaced
integers that are all the same color. Previous work by Parrilo, Robertson and
Saracino conjectured an optimal coloring for large $N$ that involves 12 colored
blocks. Here, we prove that the conjecture is optimal among anti-symmetric
colorings with 12 or fewer colored blocks. We leverage a connection to the
coloring of the continuous interval $[0,1]$ used by Parrilo, Robertson, and
Saracino as well as by Butler, Costello and Graham. Our proof identifies
classes of colorings with permutations, then counts the permutations using
mixed integer linear programming.
