Conway's thrackle conjecture: a new upper bound of $\frac{4}{3}n$

A thrackle is a drawing of a graph in the plane in which every pair of edges meets exactly once, either at a common endpoint or at a proper crossing. Conway conjectured that every thrackle has at most as many edges as vertices. We prove that every thrackle on $n\ge 4$ vertices has at most $\lfloor 4(n-1)\mathrm{/}3\rfloor$ edges. The proof combines Pammer’s computer-assisted verification of the conjecture for graphs on at most twelve vertices with a discharging argument in suitable plane and projective-plane embeddings.