# Big Ramsey degrees of 3-uniform hypergraphs

## Main Article Content

## Abstract

Given a countably infinite hypergraph $\mathcal R$ and a finite hypergraph $\mathcal A$, the \emph{big Ramsey degree} of $\mathcal A$ in $\mathcal R$ isthe least number $L$ such that, for every finite $k$ and every $k$-colouring of the embeddings of $\mathcal A$ to $\mathcal R$, there exists an embedding $f$ from $\mathcal R$ to $\mathcal R$ such that all the embeddings of $\mathcal A$ to the image $f(\mathcal R)$ 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).

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).

## Article Details

How to Cite

Balko, M., Chodounský, D., Hubička, J., Konečný, M., & Vena, L.
(2019).
Big Ramsey degrees of 3-uniform hypergraphs.

*Acta Mathematica Universitatis Comenianae, 88*(3), 415-422. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/amuc/article/view/1299/675
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Section

EUROCOMB 2019