Exact circulant Ramsey numbers beyond n=20

Determine the exact circulant Ramsey numbers R_C(3,n) for n>20 by resolving the feasibility status of every relevant graph order between the known lower and upper bounds.

Background

The paper establishes exact circulant Ramsey numbers R_C(3,n) for n=13 through 20 using a distance-space integer-programming formulation and exhaustive infeasibility certification at the relevant larger graph orders. Because circulant Ramsey colorings are not monotone in graph order, proving an exact value requires ruling out colorings at every order above the largest order known to admit one, up to a valid upper bound.

For n>20, the authors report that their computational procedure could not establish exact values because at least one graph order in the interval between the best known lower and upper bounds remained unresolved within the imposed time limit. Thus, the exact values of R_C(3,n) for these parameters are explicitly left undetermined by the study.

References

Moreover, for $R_C(3,n)$ with $n > 20$, we were not able to computationally establish the exact value of $R_C(3,n)$. This is because, for at least one graph order between $\underline{R}(3,n)$ and $\overline{R}(3,n)$, we could not verify the corresponding feasibility problem to be infeasible within the given time limit.

— An Integer Programming Approach to Compute Lower Bounds for Ramsey Numbers Using Circulant Graphs  (2608.18769 - Coniglio et al., 19 Aug 2026) in Remark following Table 4, Section 8, "Computational Study on Circulant Ramsey Numbers"