Optimal bounds for maximum degree and treewidth in online Ramsey games

Determine, for each graph parameter \(\pi\) equal to maximum degree or treewidth, number of colors \(q\ge 2\), and target bound \(d\ge 1\), the least finite value \(f_q^\pi(d)\), if it exists, such that Builder can force a monochromatic copy of every graph H with \(\pi(H)\le d\) in the q-color online Ramsey game while maintaining \(\pi(G)\le f_q^\pi(d)\) for the graph G drawn at every stage.

Background

The paper establishes that degeneracy can be preserved exactly: every d-degenerate target graph can be forced while Builder's graph remains d-degenerate. It contrasts this with maximum degree and treewidth, for which preserving the target's parameter bound is impossible in general. Existing results give lower bounds for particular targets, including maximum degree at least 2Δ−1 for forcing K_{Δ+1} and treewidth at least 3 for forcing K_3. The authors therefore leave open the problem of determining the least parameter bound sufficient for general target graphs of parameter at most d.

References

This raises the question of how large a bound is necessary. More precisely, let \pi denote either maximum degree or treewidth. For q\ge2 and d\ge1, what is the least bound f_q\pi(d), if finite, such that Builder can force a monochromatic copy of every graph H with \pi(H)\le d in the q-color online Ramsey game while keeping \pi(G)\le f_q\pi(d) for the graph G drawn at every stage?

— Tight degeneracy bounds in online Ramsey games  (2610.01516 - Chen et al., 1 Oct 2026) in Section 4, Concluding remarks