The homomorphism threshold of odd cycle $C_{2k-1}$ is below $\frac{1}{2k-1}$

The homomorphism threshold $δ_{\text{hom}}(H)$ of a graph $H$ asks how large the minimum degree of an $H$-free graph has to be in order to force a homomorphism to a bounded $H$-free graph. Determining this threshold is in general very difficult, and the odd cycles $C_{2k-1}$ are among the most impor