have chosen particular values of the $a_i$ for simplicity; indeed, we only need a single choice of coefficients to work in order to establish Theorem \[thm:main\_theorem\]. However, the same argument is able to establish Lemma \[lem:no\_bad\_cycles\], and thus also Theorem \[thm:main\_theorem\], for a much wider class of equations. For example, whenever $a_1, a_2 {\text{ and }}a_3$ are coprime, and at least two of the three coefficients are large enough, then the analogue of Lemma \[lem:no\_bad\_cycles\] holds, and thus $\gamma_{a_1, a_2, a_3} < 1/12 + {\varepsilon}$.

Conversely, it does not suffice for just one of the $a_i$ to be large. For example, if $a_1 = a_2 = 1$, then it can be shown that, for $S$ a slightly modified version of the set in Proposition \[prop:twelfth\], $T_{1, 1, a_3}(S) > \frac{1}{5} |S|^2$ for any $a_3$.

Suppose there is a cycle whose label product is not 1; consider a shortest such cycle. By minimality, such a cycle may have no repeated vertices, and thus must have at most $K_1$ vertices. Thus, without loss of generality the cycle is $S_1, S_2, \dots, S_k, S_1$, where $S_i \rightarrow S_{i+1}$ has label $t_i$ (with $S_{k+1} = S_1$), and $k {\leqslant}K_1$.

By Lemma \[lem:easy\_TNRG\_lemma\], we deduce 