TY - JOUR

T1 - Big ramsey degrees of 3-uniform hypergraphs

AU - Balko, M.

AU - Chodounský, D.

AU - Hubička, J.

AU - Konečný, M.

AU - Vena, L.

N1 - Funding Information:
Received June 7, 2019. 2010 Mathematics Subject Classification. Primary 05D10, 05C05, 05C65, 05C55, 05C80. Supported by project 18-13685Y of the Czech Science Foundation (GACˇR) and by Charles University projects Progres Q48 and GA UK No 378119.
Publisher Copyright:
© 2019, Univerzita Komenskeho. All rights reserved.

PY - 2019/9/2

Y1 - 2019/9/2

N2 - Given a countably infinite hypergraph R and affnite hypergraph A, the big Ramsey degree of A in R is the least number L such that, for every finite k and every k-colouring of the embeddings of A to R, there exists an embedding f from R to R such that all the embeddings of A to the image f(R) have at most L different colours. We describe the big Ramsey degrees of the random countably infinite 3-uniform hypergraph, thereby solving a question of Sauer. We also give a new presentation of the results of Devlin and Sauer on, respectively, big Ramsey degrees of the order of the rationals and the countably infinite random graph. Our techniques generalise (in a natural way) to relational structures and give new examples of Ramsey structures (a concept recently introduced by Zucker with applications to topological dynamics).

AB - Given a countably infinite hypergraph R and affnite hypergraph A, the big Ramsey degree of A in R is the least number L such that, for every finite k and every k-colouring of the embeddings of A to R, there exists an embedding f from R to R such that all the embeddings of A to the image f(R) have at most L different colours. We describe the big Ramsey degrees of the random countably infinite 3-uniform hypergraph, thereby solving a question of Sauer. We also give a new presentation of the results of Devlin and Sauer on, respectively, big Ramsey degrees of the order of the rationals and the countably infinite random graph. Our techniques generalise (in a natural way) to relational structures and give new examples of Ramsey structures (a concept recently introduced by Zucker with applications to topological dynamics).

UR - http://www.scopus.com/inward/record.url?scp=85073771525&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85073771525

SN - 0862-9544

VL - 88

SP - 415

EP - 422

JO - Acta Mathematica Universitatis Comenianae

JF - Acta Mathematica Universitatis Comenianae

IS - 3

ER -