Classification of crescent configurations
Let $n$ points be in crescent configuration in $\mathbb{R}^d$ if they lie in general position in $\mathbb{R}^d$ and determine $n-1$ distinct distances, such that for every $1 \leq i \leq n-1$ there is a distance that occurs exactly $i$ times. For the planar problem, Erdős conjectured that there exists $N$ such that no crescent configuration exists on $n$ points whenever $n\geq N$. In the same paper, he also gave constructions for $n$ up to 8, with $n=5$ due to Pomerance and $n=7,8$ due to Palásti, but nothing is yet known for $n \geq 9$. Most recently, Burt et al. proved that a crescent configuration on $n$ points exists in $\mathbb{R}^{n-2}$ for $n \geq 3$. In this paper, we classify planar crescent configurations by their complete distance-equality patterns, up to relabeling the vertices. We restrict attention to the strict distance classes defined below. There are three such classes on four vertices, all of which admit planar crescent realizations. On five vertices, there are $49$ strict classes, exactly $31$ of which admit planar crescent realizations. On six vertices, there are $6837$ strict classes, exactly six of which admit planar crescent realizations. The classification combines finite enumeration with exact algebraic verification of realizations and nonexistence. The distance-geometric criteria also apply in higher dimensions.