Conway's thrackle conjecture

Conway's thrackle conjecture states that every graph drawable in the plane so each pair of edges meets exactly once has at most as many edges as vertices.

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Graph theory: The mathematical study of graphs, structures made of vertices connected by edges. The conjecture bounds edges against vertices in a graph.
Thrackle: A drawing of a graph in which every pair of edges meets exactly once, at a shared endpoint or a crossing. Thrackles are the drawings to which the conjectured bound applies.
John Horton Conway: An English mathematician known for contributions to group theory, combinatorics, and recreational mathematics. Conway posed the edge-bound problem that bears his name.
Conway's thrackle conjecture bounds: Known upper and lower bounds on the number of edges in a thrackle relative to its vertices. These bounds show how far current results remain from the conjectured inequality.
Topological graph theory: The study of graphs embedded or drawn on surfaces and the properties of those embeddings. Thrackles are defined by how a graph is drawn in the plane.
Odd cycle: A cycle containing an odd number of edges. Odd cycles are a central structure in known thrackle arguments.
Paul Erdős: A Hungarian mathematician whose work shaped combinatorics, number theory, and probability. Erdős helped bring attention to the problem in combinatorial graph theory.
Linear forest: A forest in which every connected component is a path. A linear forest is a standard construction achieving the conjectured edge-to-vertex ratio.
Graph drawing: The representation of graph vertices and edges as points and curves in a geometric space. The conjecture concerns drawings, not merely abstract graph structure.
Cycle graph: A graph whose vertices form one cycle, with each vertex joined to its two neighbors. Cycles provide basic examples of graphs that can be drawn as thrackles.