Thrackle

Then, w must have degree one, because no line segment ending at w, other than vw, can touch both vx and vy.

Removing w and vw produces a smaller thrackle, without changing the difference between the numbers of edges and vertices.

[3] As Erdős also observed, the set of pairs of points realizing the diameter of a point set must form a linear thrackle: no two diameters can be disjoint from each other, because if they were then their four endpoints would have a pair at farther distance apart than the two disjoint edges.

For this reason, every set of n points in the plane can have at most n diametral pairs, answering a question posed in 1934 by Heinz Hopf and Erika Pannwitz.

[4] Andrew Vázsonyi conjectured bounds on the number of diameter pairs in higher dimensions, generalizing this problem.

An enumeration of linear thrackles may be used to solve the biggest little polygon problem, of finding an n-gon with maximum area relative to its diameter.

Conway offered a $1000 prize for proving or disproving this conjecture, as part of a set of prize problems also including Conway's 99-graph problem, the minimum spacing of Danzer sets, and the winner of Sylver coinage after the move 16.

[1][9] It has been proved that every cycle graph other than C4 has a thrackle embedding, which shows that the conjecture is sharp.