Avoiding hypergraph Ramsey theory in edge-efficient online Ramsey strategies

Determine whether the use of the hypergraph Ramsey theorem can be avoided in obtaining substantially better bounds on the number of edges drawn by Builder in the q-color online Ramsey game, while retaining the optimal degeneracy bound d for forcing any d-degenerate target graph H.

Background

The paper proves the optimal degeneracy equality f_q(d)=d, but its construction may draw a very large number of edges because it relies on the survival game theorem, hypergraph Ramsey numbers, and repeated applications of the key reduction lemma. The authors explicitly ask whether hypergraph Ramsey theory is necessary for improving these quantitative edge bounds while preserving the optimal degeneracy restriction on Builder's graph.