The maximum number of odd integral distances between points in the plane

Forbidden Subgraphs of the Odd‐Distance Graph

“…Also, since s = 2m 1,2 m 2,3 2m 2 and m 1,2 , m 2,3 = 0 (see (5)) s = 0. On the other hand, we have (see (4)): m 1,3 = 2r 1 r 3 cosθ 1,3 = 2r 1 r 3 (cos θ 1,2 cos θ 2,3 ∓ sin θ 1,2 sin θ 2,3 ).…”

“…If there exists a representation of W 5 in R 2 such that some m i,i+2 are rational and some are irrational, then there exists a representation of W 5 such that all rational m i,i+2 will be integers. 5 be the vertices of W 5 embedded in R 2 and m i, j be its corresponding entries in the matrix M (see (4)).…”

“…Thus started the pursuit of unveiling the mysteries of , whose “close cousin,” the unit‐distance graph, has been haunting mathematicians since its introduction in 1950. Since every finite subgraph of is K 4 ‐free, it follows from Turán's theorem that the maximum number of odd distances among n points in the plane is , the number of edges in the complete tripartite graph on n vertices whose three partitions are “as equal as possible.” Piepemeyer proved that this maximum number of odd distances is attained by showing that the complete tripartite graph (and therefore any ) is a subgraph of .…”

The Odd-Distance Plane Graph

“…On the other hand, the only forbidden subgraph of the odddistance graph known to us is K 4 . Finding the maximum number of edges among all subgraphs of order n is unknown for the unit-distance graph but follows a very predictive upper bound for the odd-distance graph as proved in [8]. It is interesting to note that P. Erdős formulated this question for the unit-distance graph 4 years before Nelson, but not in the context of graphs, see [3].…”

“…Since the odd-distance graph spanned by n points does not contain K 4 as a subgraph, this number is bounded by Turán's function. L. Piepmeyer [8] showed that the complete tri-partite graph K m,m,m can be embedded in the plane (actually on a circle) in such a way that two vertices connected by an edge will be at odd distance apart. Note that this is a faithful embedding (no other vertices are at odd distance apart).…”