Conway's Thrackle Conjecture

A thrackle is a drawing of a graph in the plane where every pair of edges meets exactly once (shared endpoint, or proper crossing). Claim: every thrackle has at most as many edges as vertices (|E| ≤ |V|).

Conway popularised it circa 1969 (Woodall noted earlier forms) and offered $1,000. The bound has been ground down: from trivialities through Cain's 1.5|V| to Fulek and Tótoh's current |E| ≤ 1.3984|V|. Every straight-line thrackle satisfies the bound (a theorem), so only genuinely curvy drawings resist. Fifty-plus years of topological graph theory grew around this deceptively tiny question.