Ordered Ramsey numbers of 3-uniform hypergraphs with bounded weak degeneracy

@misc{chen2026ordered,
  title = {{Ordered Ramsey numbers of 3-uniform hypergraphs with bounded weak degeneracy}},
  author = {Wen Chen and Zihan He and Qizhong Lin and Meng Liu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16767},
  eprint = {2609.16767},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16767},
}

TY  - UNPB
TI  - Ordered Ramsey numbers of 3-uniform hypergraphs with bounded weak degeneracy
AU  - Wen Chen
AU  - Zihan He
AU  - Qizhong Lin
AU  - Meng Liu
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.16767
UR  - https://arxiv.org/abs/2609.16767
AB  - The \emph{ordered Ramsey number} r_<(G,H) of ordered k-graphs G and H is the least integer N such that every red-blue edge-coloring of the naturally ordered complete k-graph on [N] contains a blue ordered copy of G or a red ordered copy of H. We prove that there is an absolute constant c>0 such that, for every integer d≥1, there is a constant C_d>0 for which every weakly d-degenerate ordered $3$-graph H on t vertices satisfies r_<(H,K_3^(3)(n)) ≤ t 2^C_d n^2−c/d for every positive integer n. This resolves a problem posed by Balko and Vizer ({\em SIAM J. Discrete Math., 2022}) in a stronger form. Furthermore, we show that the weak-degeneracy hypothesis cannot be replaced by bounded standard degeneracy. In particular, for every sufficiently large n, there exists a $1$-degenerate ordered $3$-graph F on at most 2^O(n) vertices such that r_<(F,K_3^(3)(n))>2^Ω(n^2).
ER  -