Spoiler for agree with rainman

with 4 colors it would definitely be 1/2.

the goal is to gaurantee 2 points of the same color are that distance apart.

so finding a construction where 1/2 works is not a solution, you have to prove that you can always get 1/2, no matter how you color it with 4 colors.

so we need to find the worst case scenario.

as rainman pointed out each of the midpoints of the triangle are 1/2 distance apart.

since we must cover all three midpoints with some color, and coloring any two midpoints the same color would allow for 1/2, the worst we can we can do is have all 3 midpoints a different color. but then we also need the vertexes of the triangle to be a diffent color than the midpoint to try to prevent 1/2. since we only have 4 colors, and six points to worry about to prevent 1/2, 1/2 is the answer.

with 4 colors it would definitely be 1/2.

the goal is to gaurantee 2 points of the same color are that distance apart.

so finding a construction where 1/2 works is not a solution, you have to prove that you can always get 1/2, no matter how you color it with 4 colors.

so we need to find the worst case scenario.

as rainman pointed out each of the midpoints of the triangle are 1/2 distance apart.

since we must cover all three midpoints with some color, and coloring any two midpoints the same color would allow for 1/2, the worst we can we can do is have all 3 midpoints a different color. but then we also need the vertexes of the triangle to be a diffent color than the midpoint to try to prevent 1/2. since we only have 4 colors, and six points to worry about to prevent 1/2, 1/2 is the answer.