Ramsey numbers for cycles versus the eleven-vertex wheel in the remaining range

Determine the exact values of the Ramsey numbers R(C_n,W_{11}) for n\in\{10,11,12,13\}, where W_{11}=K_1+C_{10}.

Background

The paper proves R(C_n,W_{11})=2n-1 for every n\ge14, extending previously established results for n\ge16. The four cases n=10,11,12,13 remain unresolved. The authors note that their Hamilton-connected core lemma applies throughout this smaller range, but the available degree bounds do not by themselves provide the connectivity and prescribed-path structure needed to complete the Ramsey argument.

References

The present arguments do not settle $R(C_n,W_{11})$ for $10\le n\le13$. The parameter range of Lemma~\ref{lem:core} already includes all four cases, since $k=5$ requires only $n\ge10$. The remaining difficulty is to force a suitable core or to construct paths of the required orders with prescribed endpoints.

— Hamilton-connected cores and five cycle--wheel Ramsey numbers  (2609.39508 - Shao et al., 30 Sep 2026) in Section “Concluding remarks”