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Since ancient times mathematicians consider geometrical objects with integralside lengths. We consider plane integral point sets $\mathcal{P}$, which aresets of $n$ points in the plane with pairwise integral distances where not allthe points are collinear.  The largest occurring distance is called its diameter. Naturally the questionabout the minimum possible diameter $d(2,n)$ of a plane integral point setconsisting of $n$ points arises. We give some new exact values and describestate-of-the-art algorithms to obtain them. It turns out that plane integralpoint sets with minimum diameter consist very likely of subsets with manycollinear points. For this special kind of point sets we prove a lower boundfor $d(2,n)$ achieving the known upper bound $n^{c_2\log\log n}$ up to aconstant in the exponent.  A famous question of Erd\H{o}s asks for plane integral point sets with no 3points on a line and no 4 points on a circle. Here, we talk of point sets ingeneral position and denote the corresponding minimum diameter by$\dot{d}(2,n)$. Recently $\dot{d}(2,7)=22 270$ could be determined via anexhaustive search.

On the minimum diameter of plane integral point sets