Hostname: page-component-7bb8b95d7b-qxsvm Total loading time: 0 Render date: 2024-09-22T02:01:18.424Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Ramsey numbers of daisies II

Part of: Extremal combinatorics Graph theory

Published online by Cambridge University Press:  18 September 2024

Marcelo Sales Open the ORCID record for Marcelo Sales [Opens in a new window]
Marcelo Sales*
Affiliation:
Department of Mathematics, University of California, Irvine, CA, USA
*
Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A $(k+r)$-uniform hypergraph $H$ on $(k+m)$ vertices is an $(r,m,k)$-daisy if there exists a partition of the vertices $V(H)=K\cup M$ with $|K|=k$, $|M|=m$ such that the set of edges of $H$ is all the $(k+r)$-tuples $K\cup P$, where $P$ is an $r$-tuple of $M$. We obtain an $(r-2)$-iterated exponential lower bound to the Ramsey number of an $(r,m,k)$-daisy for $2$-colours. This matches the order of magnitude of the best lower bounds for the Ramsey number of a complete $r$-graph.

Keywords

Ramsey theoryhypergraphsstepping-up lemma

MSC classification

Primary: 05D10: Ramsey theory
Secondary: 05C65: Hypergraphs
Type
Paper
Information
Combinatorics, Probability and Computing , First View , pp. 1 - 27
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

*

The author was supported by NSF grant DMS 1764385 and US Air Force grant FA9550-23-1-0298.

References

Bollobás, B., Leader, I. and Malvenuto, C. (2011) Daisies and other Turán problems. Combin. Probab. Comput. 20(5) 743–747.CrossRefGoogle Scholar
Conlon, D., Fox, J. and Sudakov, B. (2013) An improved bound for the stepping-up lemma. Discrete Appl. Math. 161(9) 1191–1196.CrossRefGoogle Scholar
Erdős, P., Hajnal, A. and Rado, R. (1965) Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 93–196.CrossRefGoogle Scholar
Graham, R. L., Rothschild, B. L. and Spencer, J. H. (2013) Ramsey Theory, Wiley Series in Discrete Mathematics and Optimization. Hoboken, NJ: John Wiley & Sons, Inc., Paperback edition of the second, 1990 edition [MR1044995].Google Scholar
Pudlák, P., Rödl, V. and Sales, M. (2024) On the Ramsey number of daisies I. doi: 10.1017/S0963548324000221.CrossRefGoogle Scholar