Polynomial edge-ordered Ramsey bound for degenerate graphs

Prove that every edge-ordered d-degenerate graph H^< on n vertices satisfies R_\prec(H^<)≤n^{O(d)}.

Background

The known upper bound for an edge-ordered d-degenerate graph is n{600d3 log(d+1)}. The conjecture proposes reducing the dependence on d to a linear factor in the exponent.

References

For sparser edge-ordered graphs, Fox and Li conjectured that the upper bound from Theorem~\ref{thm-ordRamHyper-foxLiDegen} can be improved.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Conjecture~\ref{prob-ordRamHyper-edgeDegen}, Section “Edge-ordered Ramsey Numbers”

For sparser edge-ordered graphs, Fox and Li conjectured that the upper bound from Theorem~\ref{thm-ordRamHyper-foxLiDegen} can be improved.

If $H\prec$ is an edge-ordered $d$-degenerate graph on $n$ vertices, then $R_\prec(H\prec) \leq n{O(d)}$.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Conjecture prob-ordRamHyper-edgeDegen, Section 4

If $H\prec$ is an edge-ordered $d$-degenerate graph on $n$ vertices, then $R_\prec(H\prec) \leq n{O(d)}$.

— A Survey on Ordered Ramsey Numbers  (2502.02155 - Balko, 4 Feb 2025) in Conjecture prob-ordRamHyper-edgeDegen, Section “Edge-ordered Ramsey Numbers”