# The largest possible cardinality of a subset of the Euclidean plane under specific conditions

Let’s define A as a subset of the Euclidean plane, such that:

1. No three points of A are collinear
2. Distance between any two points of A is rational

What is the largest possible cardinality of A?

There exist quite many such sets with cardinality 4, but I failed to construct anything larger.

Any help will be appreciated.

Let $\alpha$ be the angle such that $\cos\alpha=3/5,\sin\alpha=4/5$. Consider the points $P_n(\sin(2^n\alpha),\cos(2^n\alpha))$ for $n\geq 1$, all lying on the unit circle. By looking at the triangle formed by $(0,0),P_n,P_m$ we find that the distance between $P_n$ and $P_m$ is $$2\sin\left(\frac{2^m-2^n}{2}\alpha\right)$$ which (trigonometry exercise) is rational because of our choice of $\alpha$. This gives a countable set as in question (since no three points on a circle are collinear).
To prove there is no larger one, consider any two points $P,Q$ in our set. For any pair of rational numbers $a,b$ there are at most two points in the plane at distance $a$ from $P$ and at distance $b$ from $Q$. Hence there can only be countably many other points in this set. Similar argument (starting with points $P,Q,R$ or so) shows that there are no uncountable such sets even in the Euclidean 3D space, nor are there in higher dimensional ones.
• Cool! ${}{}{}{}$ – rschwieb Aug 9 '17 at 21:40