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on Conway thrackle constant ($C_{78}$)

The statement

In topological graph theory, a thrackle is a drawing of a finite graph in the plane in which every pair of edges meets precisely once, either at a common endpoint or at a proper crossing. If $t(n)$ denotes the maximum number of edges in a thrackle on $n$ vertices, then Conway's thrackle conjecture states that $t(n)=n$ for every $n\ge 3$ . [FP2011-def-tn-conj]

For a finite graph $G$ , write $V(G)$ for its vertex set and $E(G)$ for its edge set. We define

C_{78} \ :=\ \sup\left\{ \frac{\lvert E(G)\rvert}{\lvert V(G)\rvert} : G \text{ admits a thrackle drawing in the plane} \right\}.

Equivalently,

C_{78} \ =\ \sup_{n\ge 1}\frac{t(n)}{n}.

Thus Conway's thrackle conjecture is equivalent to the assertion that $C_{78}=1$ . [FP2011-def-tn-conj]

At present, the best established range is

1\ \le\ C_{78}\ \le\ 1.393.

The lower bound comes from thrackled cycles, and the upper bound from Xu's current record bound $m\le 1.393(n-1)$ as recorded in recent survey-style sources. [FP2011-open-sharp] [KSTZ2025-current-best-special]

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.