Vertex degree-Ramsey boundedness for all graphs

Determine whether the vertex degree-Ramsey number R^v_\Delta(G;s) is bounded by a function depending only on the maximum degree \Delta(G) and the number of colors s, for every finite graph G and every positive integer s.

Background

The paper introduces the vertex degree-Ramsey number Rv_\Delta(G;s), defined using vertex colorings of a host graph and requiring a monochromatic copy of the target graph G. A graph family is vertex-R_\Delta-bounded if these numbers are bounded solely in terms of the target's maximum degree and the number of colors.

The authors prove positive results for subdivisions of bounded-degree graphs, trees, closed blowups of trees, and bounded-treewidth graphs, but explicitly state that these results do not resolve the corresponding question for arbitrary graphs themselves. Thus, the general vertex degree-Ramsey boundedness problem remains unresolved.

References

While we are not able to answer Question~\ref{qn:vertex-main}, our main result gives a positive answer once the required monochromatic target structure is enlarged from $G$ itself to the collection of subdivisions of $G$.

— Forcing monochromatic subdivisions  (2609.00636 - Collado et al., 1 Sep 2026) in Section 1, Introduction; Question 1.2 (labeled Question~\ref{qn:vertex-main})