An integral set of points in the plane is one in which the distance between any pairs of points is an integer. Anning and Erdos proved that the points in an infinite integral set must be a collinear.
If we add the condition that the points of an integral set be in general position (no three on a line, no four on a circle), then it is not known how large such a set can be. In fact, it is not even known whether a set of size 7 exists.
In any case, it appears that if large integral sets exist, their diameter grows rapidly as a function of their size. To investigate this growth, we add the further requirement that the points in our sets have integer coordinates. A (3,4,5) right triangle gives an integral set of size 3 with integer coordinates and diameter 4, clearly the smallest possible. It is almost as easy to see that the smallest set of size 4 is the one shown here, and whose diameter is 8.
For sets of size 5, things become a little more challenging. The smallest such sets have diameter 78 and one is shown here. Finally, smallest sets of size 6 have diameter 1886. Such a set is shown here.
Note that if we omit the requirement that no 4 points can be on a circle, then it is much easier to find large sets. For example, here is a set of size 8 and here is a set of size 12.
Geoff Exoo
