Abstract
A theorem of Erdős asserts that every infinite \({X \subseteq \mathbb{R}^n}\) has a subset of the same cardinality having no repeated distances. This theorem is generalized here as follows: If \({(\mathbb{R}^n,E)}\) is an algebraic hypergraph that does not have an infinite, complete subset, then every infinite subset has an independent subset of the same cardinality.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Davies Roy O.: Partitioning the plane into denumberably many sets without repeated distances. Proc. Cambridge Philos. Soc. 72, 179–183 (1972)
Erdős P.: Some remarks on set theory. Proc. Amer. Math. Soc. 1, 127–141 (1950)
P. Erdős, Set-theoretic, measure-theoretic, combinatorial, and number-theoretic problems concerning point sets in Euclidean space, Real Anal. Exchange, 4 (1978/79), 113–138.
Kunen Kenneth.: Partitioning Euclidean space. Math. Proc. Cambridge Philos. Soc. 102, 379–383 (1987)
Schmerl J.H.: Countable partitions of Euclidean space. Math. Proc. Cambridge Philos. Soc. 120, 7–12 (1996)
Schmerl J.H.: Avoidable algebraic subsets of Euclidean space. Trans. Amer. Math. Soc. 352, 2479–2489 (2000)
Schmerl J.H.: Chromatic numbers of algebraic hypergraphs. Combinatorica 37, 1011–1026 (2017)
J. H. Schmerl, Deciding the chromatic numbers of algebraic hypergraphs, J. Symb. Logic (to appear).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Schmerl, J.H. A remark on a theorem of Erdős. Acta Math. Hungar. 155, 489–498 (2018). https://doi.org/10.1007/s10474-018-0830-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-018-0830-y