A General Upper Bound on Multicolor Ordered Ramsey Numbers

@misc{balko2026general,
  title = {{A General Upper Bound on Multicolor Ordered Ramsey Numbers}},
  author = {Martin Balko and Klára Grinerová},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28075},
  eprint = {2609.28075},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28075},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - A General Upper Bound on Multicolor Ordered Ramsey Numbers
AU  - Martin Balko
AU  - Klára Grinerová
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28075
UR  - https://arxiv.org/abs/2609.28075
AB  - We provide a general upper bound on multicolor ordered Ramsey numbers in terms of the interval chromatic number and the degeneracy of an ordered graph. We extend previous results by Conlon, Fox, Lee, and Sudakov (2017) by showing that for every n-vertex ordered graph G^< with degeneracy d≥2, and interval chromatic number χ, its q-color ordered Ramsey number satisfies r_<(G^<;q) ∈ n^O(d^q−1⌈ logχ⌉^q−1) for every q ≥ 2. For fixed parameters q,d,χ, the resulting estimate is polynomial in n. For triangle-free ordered graphs G^<, we also provide the stronger estimate n^O(q^2d ⌈ log χ⌉^q−1). It also follows from a recent result by Li (2026) that our upper bound is almost tight for ordered matchings.
ER  -