Zero-sum Ramsey numbers of double-stars

Fully determine the zero-sum Ramsey number R(T,Z_3) for every double-star T=D(k_1,k_2) having a number of edges divisible by 3.

Background

The paper identifies double-stars D(k_1,k_2), with k_1≥k_2≥2, as a family for which no upper bound better than n+2 is known. It notes that the subfamily D(k_1,1) is already resolved by an earlier corollary, while determining the values for all double-stars with 0 modulo 3 edges would provide progress toward the broader tree conjecture.

References

Fully determine $R(T, \mathbb{Z}_3)$ for the family of double-stars on $0 \ (!!!!!!!!\mod 3)$ edges.

— On zero-sum Ramsey numbers modulo 3  (2502.03864 - Caro et al., 6 Feb 2025) in Section 5, “Concluding remarks,” Problem