On edge-ordered Ramsey numbers of hypergraphs
Abstract
For an integer h≥2, an edge-ordered h-uniform hypergraph H< is an h-uniform hypergraph with a linear ordering of its hyperedges. For an integer q ≥ 1, the edge-ordered Ramsey number Redge(H<;q) of H< is the smallest positive integer N such that there exists an edge-ordered h-uniform hypergraph G< on N vertices such that, for every q-coloring of the hyperedges of G<, there is a monochromatic copy of H< in G< preserving the hyperedge order. In this note, we show that, for every edge-ordered h-uniform hypergraph F< on n vertices with m≥1 hyperedges, Redge(F<;q) ≤ Rh(F;q)Cm2 where F is the underlying unordered hypergraph, Rh(F;q) is its standard q-color Ramsey number, and C=C(h,q)>0 is a constant. For fixed h≥3 and q≥2, this shows that prescribing the hyperedge order does not require an increase in the tower height of the classical upper bound. It also settles a problem posed by Fox and Li (2020), and for h=2 gives the bound 2O(n4log n), improving their bounds for every fixed q ≥ 3.
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BibTeX
@misc{balko2026edgeordered,
title = {{On edge-ordered Ramsey numbers of hypergraphs}},
author = {Martin Balko and Jelena Glišić},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39653},
eprint = {2610.39653},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39653},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On edge-ordered Ramsey numbers of hypergraphs
AU - Martin Balko
AU - Jelena Glišić
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39653
UR - https://arxiv.org/abs/2610.39653
AB - For an integer h≥2, an edge-ordered h-uniform hypergraph H^< is an h-uniform hypergraph with a linear ordering of its hyperedges. For an integer q ≥ 1, the edge-ordered Ramsey number R_edge(H^<;q) of H^< is the smallest positive integer N such that there exists an edge-ordered h-uniform hypergraph G^< on N vertices such that, for every q-coloring of the hyperedges of G^<, there is a monochromatic copy of H^< in G^< preserving the hyperedge order. In this note, we show that, for every edge-ordered h-uniform hypergraph F^< on n vertices with m≥1 hyperedges, R_edge(F^<;q) ≤ R_h(F;q)^Cm^2 where F is the underlying unordered hypergraph, R_h(F;q) is its standard q-color Ramsey number, and C=C(h,q)>0 is a constant. For fixed h≥3 and q≥2, this shows that prescribing the hyperedge order does not require an increase in the tower height of the classical upper bound. It also settles a problem posed by Fox and Li (2020), and for h=2 gives the bound 2^O(n^4log n), improving their bounds for every fixed q ≥ 3.
ER -