Type-respecting amalgamation and big Ramsey degrees

EUROCOMB’23

Abstract
We give an infinitary extension of the Nešetřil-Rödl theorem for category of relational structures with special type-respecting embeddings.

Pages:
57–65
References

Martin Balko, David Chodounský, Natasha Dobrinen, Jan Hubička, Matěj Konečný, Lluis Vena, and Andy Zucker. Exact big Ramsey degrees via coding trees. ArXiv:2110.08409, 2021.

Martin Balko, David Chodounský, Natasha Dobrinen, Jan Hubička, Matěj Konečný, Lluis Vena, and Andy Zucker. Characterisation of the big Ramsey degrees of the generic partial order. ArXiv:2303.10088, 2023.

Martin Balko, David Chodounský, Natasha Dobrinen, Jan Hubička, Jaroslav Nešetřil, Matěj Konečný, and Andy Zucker. Ramsey theorem for trees with successor operation. In preparation, 2023.

Martin Balko, David Chodounský, Jan Hubička, Matěj Konečný, and Lluis Vena. Big Ramsey degrees of 3-uniform hypergraphs. Acta Mathematica Universitatis Comenianae, 88(3):415-422, 2019. Extended abstract for Eurocomb 2021.

Martin Balko, David Chodounský, Jan Hubička, Matěj Konečný, and Lluis Vena. Big Ramsey degrees of 3-uniform hypergraphs are finite. Combinatorica, 42(2):659-672, 2022.
https://doi.org/10.1007/s00493-021-4664-9

Samuel Braunfeld, David Chodounský, Noé de Rancourt, Jan Hubička, Jamal Kawach, and Matěj Konečný. Big Ramsey degrees and infinite languages. ArXiv:2301.13116, 2023.

Timothy J. Carlson and Stephen G. Simpson. A dual form of Ramsey's theorem. Advances in Mathematics, 53(3):265-290, 1984.
https://doi.org/10.1016/0001-8708(84)90026-4

Denis Devlin. Some partition theorems and ultrafilters on ω. PhD thesis, Dartmouth College, 1979.

Natasha Dobrinen. The Ramsey theory of the universal homogeneous triangle-free graph. Journal of Mathematical Logic, 20(02):2050012, 2020.
https://doi.org/10.1142/S0219061320500129

Natasha Dobrinen. Ramsey theory of homogeneous structures: current trends and open problems. arXiv preprint arXiv:2110.00655; Survey for ICM'22, 2021.

Natasha Dobrinen. The Ramsey theory of Henson graphs. Journal of Mathematical Logic, 0(0):2250018, 2023.

Alfred W. Hales and Robert I. Jewett. Regularity and positional games. Transactions of the American Mathematical Society, 106:222-229, 1963.
https://doi.org/10.1090/S0002-9947-1963-0143712-1

James D. Halpern and Hans Läuchli. A partition theorem. Transactions of the American Mathematical Society, 124(2):360-367, 1966.
https://doi.org/10.1090/S0002-9947-1966-0200172-2

Wilfrid Hodges. Model theory, volume 42. Cambridge University Press, 1993.
https://doi.org/10.1017/CBO9780511551574

Jan Hubička. Big Ramsey degrees using parameter spaces. ArXiv:2009.00967, 2020.

Keith R. Milliken. A Ramsey theorem for trees. Journal of Combinatorial Theory, Series A, 26(3):215-237, 1979.
https://doi.org/10.1016/0097-3165(79)90101-8

Jaroslav Nešetřil and Vojtěch Rödl. A structural generalization of the Ramsey theorem. Bulletin of the American Mathematical Society, 83(1):127-128, 1977.
https://doi.org/10.1090/S0002-9904-1977-14212-2

Jaroslav Nešetřil and Vojtěch Rödl. The partite construction and Ramsey set systems. Discrete Mathematics, 75(1):327-334, 1989.
https://doi.org/10.1016/0012-365X(89)90097-6

Norbert Sauer. A ramsey theorem for countable homogeneous directed graphs. Discrete Mathematics, 253:45-61, 06 2002.
https://doi.org/10.1016/S0012-365X(01)00448-4

Stevo Todorcevic. Introduction to Ramsey spaces, volume 174. Princeton University Press, 2010.
https://doi.org/10.1515/9781400835409

Andy Zucker. On big Ramsey degrees for binary free amalgamation classes. Advances in Mathematics, 408:108585, 2022. Martin Balko, David Chodounský, Natasha Dobrinen, Jan Hubička, Matěj Konečný, Lluis Vena, and Andy Zucker. Exact big Ramsey degrees via coding trees. ArXiv:2110.08409, 2021.
https://doi.org/10.1016/j.aim.2022.108585

Martin Balko, David Chodounský, Natasha Dobrinen, Jan Hubička, Matěj Konečný, Lluis Vena, and Andy Zucker. Characterisation of the big Ramsey degrees of the generic partial order. ArXiv:2303.10088, 2023.

Martin Balko, David Chodounský, Natasha Dobrinen, Jan Hubička, Jaroslav Nešetřil, Matěj Konečný, and Andy Zucker. Ramsey theorem for trees with successor operation. In preparation, 2023.

Martin Balko, David Chodounský, Jan Hubička, Matěj Konečný, and Lluis Vena. Big Ramsey degrees of 3-uniform hypergraphs. Acta Mathematica Universitatis Comenianae, 88(3):415-422, 2019. Extended abstract for Eurocomb 2021.

Martin Balko, David Chodounský, Jan Hubička, Matěj Konečný, and Lluis Vena. Big Ramsey degrees of 3-uniform hypergraphs are finite. Combinatorica, 42(2):659-672, 2022.
https://doi.org/10.1007/s00493-021-4664-9

Samuel Braunfeld, David Chodounský, Noé de Rancourt, Jan Hubička, Jamal Kawach, and Matěj Konečný. Big Ramsey degrees and infinite languages. ArXiv:2301.13116, 2023.

Timothy J. Carlson and Stephen G. Simpson. A dual form of Ramsey's theorem. Advances in Mathematics, 53(3):265-290, 1984.
https://doi.org/10.1016/0001-8708(84)90026-4

Denis Devlin. Some partition theorems and ultrafilters on ω. PhD thesis, Dartmouth College, 1979.

Natasha Dobrinen. The Ramsey theory of the universal homogeneous triangle-free graph. Journal of Mathematical Logic, 20(02):2050012, 2020.
https://doi.org/10.1142/S0219061320500129

Natasha Dobrinen. Ramsey theory of homogeneous structures: current trends and open problems. arXiv preprint arXiv:2110.00655; Survey for ICM'22, 2021.

Natasha Dobrinen. The Ramsey theory of Henson graphs. Journal of Mathematical Logic, 0(0):2250018, 2023.

Alfred W. Hales and Robert I. Jewett. Regularity and positional games. Transactions of the American Mathematical Society, 106:222-229, 1963.
https://doi.org/10.1090/S0002-9947-1963-0143712-1

James D. Halpern and Hans Läuchli. A partition theorem. Transactions of the American Mathematical Society, 124(2):360-367, 1966.
https://doi.org/10.1090/S0002-9947-1966-0200172-2

Wilfrid Hodges. Model theory, volume 42. Cambridge University Press, 1993.
https://doi.org/10.1017/CBO9780511551574

Jan Hubička. Big Ramsey degrees using parameter spaces. ArXiv:2009.00967, 2020.

Keith R. Milliken. A Ramsey theorem for trees. Journal of Combinatorial Theory, Series A, 26(3):215-237, 1979.
https://doi.org/10.1016/0097-3165(79)90101-8

Jaroslav Nešetřil and Vojtěch Rödl. A structural generalization of the Ramsey theorem. Bulletin of the American Mathematical Society, 83(1):127-128, 1977.
https://doi.org/10.1090/S0002-9904-1977-14212-2

Jaroslav Nešetřil and Vojtěch Rödl. The partite construction and Ramsey set systems. Discrete Mathematics, 75(1):327-334, 1989.
https://doi.org/10.1016/0012-365X(89)90097-6

Norbert Sauer. A ramsey theorem for countable homogeneous directed graphs. Discrete Mathematics, 253:45-61, 06 2002.
https://doi.org/10.1016/S0012-365X(01)00448-4

Stevo Todorcevic. Introduction to Ramsey spaces, volume 174. Princeton University Press, 2010.
https://doi.org/10.1515/9781400835409

Andy Zucker. On big Ramsey degrees for binary free amalgamation classes. Advances in Mathematics, 408:108585, 2022. Martin Balko, David Chodounský, Natasha Dobrinen, Jan Hubička, Matěj Konečný, Lluis Vena, and Andy Zucker. Exact big Ramsey degrees via coding trees. ArXiv:2110.08409, 2021.
https://doi.org/10.1016/j.aim.2022.108585

Martin Balko, David Chodounský, Natasha Dobrinen, Jan Hubička, Matěj Konečný, Lluis Vena, and Andy Zucker. Characterisation of the big Ramsey degrees of the generic partial order. ArXiv:2303.10088, 2023.

Martin Balko, David Chodounský, Natasha Dobrinen, Jan Hubička, Jaroslav Nešetřil, Matěj Konečný, and Andy Zucker. Ramsey theorem for trees with successor operation. In preparation, 2023.

Martin Balko, David Chodounský, Jan Hubička, Matěj Konečný, and Lluis Vena. Big Ramsey degrees of 3-uniform hypergraphs. Acta Mathematica Universitatis Comenianae, 88(3):415-422, 2019. Extended abstract for Eurocomb 2021.

Martin Balko, David Chodounský, Jan Hubička, Matěj Konečný, and Lluis Vena. Big Ramsey degrees of 3-uniform hypergraphs are finite. Combinatorica, 42(2):659-672, 2022.
https://doi.org/10.1007/s00493-021-4664-9

Samuel Braunfeld, David Chodounský, Noé de Rancourt, Jan Hubička, Jamal Kawach, and Matěj Konečný. Big Ramsey degrees and infinite languages. ArXiv:2301.13116, 2023.

Timothy J. Carlson and Stephen G. Simpson. A dual form of Ramsey's theorem. Advances in Mathematics, 53(3):265-290, 1984.
https://doi.org/10.1016/0001-8708(84)90026-4

Denis Devlin. Some partition theorems and ultrafilters on ω. PhD thesis, Dartmouth College, 1979.

Natasha Dobrinen. The Ramsey theory of the universal homogeneous triangle-free graph. Journal of Mathematical Logic, 20(02):2050012, 2020.
https://doi.org/10.1142/S0219061320500129

Natasha Dobrinen. Ramsey theory of homogeneous structures: current trends and open problems. arXiv preprint arXiv:2110.00655; Survey for ICM'22, 2021.

Natasha Dobrinen. The Ramsey theory of Henson graphs. Journal of Mathematical Logic, 0(0):2250018, 2023.

Alfred W. Hales and Robert I. Jewett. Regularity and positional games. Transactions of the American Mathematical Society, 106:222-229, 1963.
https://doi.org/10.1090/S0002-9947-1963-0143712-1

James D. Halpern and Hans Läuchli. A partition theorem. Transactions of the American Mathematical Society, 124(2):360-367, 1966.
https://doi.org/10.1090/S0002-9947-1966-0200172-2

Wilfrid Hodges. Model theory, volume 42. Cambridge University Press, 1993.
https://doi.org/10.1017/CBO9780511551574

Jan Hubička. Big Ramsey degrees using parameter spaces. ArXiv:2009.00967, 2020.

Keith R. Milliken. A Ramsey theorem for trees. Journal of Combinatorial Theory, Series A, 26(3):215-237, 1979.
https://doi.org/10.1016/0097-3165(79)90101-8

Jaroslav Nešetřil and Vojtěch Rödl. A structural generalization of the Ramsey theorem. Bulletin of the American Mathematical Society, 83(1):127-128, 1977.
https://doi.org/10.1090/S0002-9904-1977-14212-2

Jaroslav Nešetřil and Vojtěch Rödl. The partite construction and Ramsey set systems. Discrete Mathematics, 75(1):327-334, 1989.
https://doi.org/10.1016/0012-365X(89)90097-6

Norbert Sauer. A ramsey theorem for countable homogeneous directed graphs. Discrete Mathematics, 253:45-61, 06 2002.
https://doi.org/10.1016/S0012-365X(01)00448-4

Stevo Todorcevic. Introduction to Ramsey spaces, volume 174. Princeton University Press, 2010.
https://doi.org/10.1515/9781400835409

Andy Zucker. On big Ramsey degrees for binary free amalgamation classes. Advances in Mathematics, 408:108585, 2022.
https://doi.org/10.1016/j.aim.2022.108585

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