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Big Ramsey degrees and structures in relational languages

  • Yifan Hu
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Abstract

We establish a precompact transfer theorem for big Ramsey structures and a uniform envelope theorem for valuation hypergraphs. These yield big Ramsey structures for generic injective relational structures in languages with finitely many symbols of each positive arity, answering the existence part of Problem~6.3.2 of Balko et al.\ [M.~Balko et al., J.~Eur.~Math.~Soc.\ 28, 2101--2150 (2026)], and refute Conjecture~4.7 of Braunfeld et al.\ [S.~Braunfeld et al., Adv.~Comb., Paper No.~4 (2024)]. We also construct a finitely constrained free amalgamation class in a finite language of maximum arity three whose Fraïssé limit has infinite singleton big Ramsey degree, giving a negative answer to the existence question in Problem~6.3.3 of Balko et al.

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Cite

BibTeXRIS
BibTeX
@misc{hu2026ramsey,
  title = {{Big Ramsey degrees and structures in relational languages}},
  author = {Yifan Hu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25715},
  eprint = {2610.25715},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25715},
}
RIS
TY  - UNPB
TI  - Big Ramsey degrees and structures in relational languages
AU  - Yifan Hu
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.25715
UR  - https://arxiv.org/abs/2610.25715
AB  - We establish a precompact transfer theorem for big Ramsey structures and a uniform envelope theorem for valuation hypergraphs. These yield big Ramsey structures for generic injective relational structures in languages with finitely many symbols of each positive arity, answering the existence part of Problem~6.3.2 of Balko et al.\ [M.~Balko et al., J.~Eur.~Math.~Soc.\ 28, 2101--2150 (2026)], and refute Conjecture~4.7 of Braunfeld et al.\ [S.~Braunfeld et al., Adv.~Comb., Paper No.~4 (2024)]. We also construct a finitely constrained free amalgamation class in a finite language of maximum arity three whose Fraïssé limit has infinite singleton big Ramsey degree, giving a negative answer to the existence question in Problem~6.3.3 of Balko et al.
ER  -