For every pair of integers q≥ 2 and d≥ 1, and every d-degenerate graph H, does Builder have a strategy in the q-color online Ramsey game that forces Painter to produce a monochromatic copy of H while the graph exposed by Builder has degeneracy at most d? Equivalently, is the known lower bound d on the degeneracy required to force every d-degenerate graph sharp for all q≥ 2 and d≥ 1?

Conlon–Fox–Sudakov online Ramsey degeneracy problem

The problem asks whether the known lower bound in the $q$ -color online Ramsey game is sharp for every $q \ge 2$ and $d \ge 1$ . The earlier literature established matching bounds only in special cases and left the general sharpness question open.

Known results

Builder can force a monochromatic copy of every $d$ -degenerate graph while exposing a graph of degeneracy at most $qd-(q-1)$ .
Painter necessarily produces a graph of degeneracy at least $d$ .
The bounds match when $d=1$ ; the general sharpness question was explicitly left open.

October 2026 claimed resolution

Wen Chen, Qizhong Lin, and Shixi Song claim in an unrefereed preprint that both the upper construction and the matching lower bound hold for all $q \ge 2$ and $d \ge 1$ , which would settle the exact degeneracy problem. No independent proof assessment, correction, or verification was found.

Current status (as of October 2026): The general problem is claimed solved by the Chen–Lin–Song preprint, but that claim remains unverified; the earlier bounds and the $d=1$ case are established.

Conlon–Fox–Sudakov online Ramsey degeneracy problem

Primary source

Additional references

Progress summary

Solutions 0