A Remark on a Theorem of Erdos

TL;DR

This paper generalizes Erdos's theorem on infinite subsets of Euclidean space, showing that in certain algebraic hypergraphs without infinite complete subsets, every infinite subset contains a large independent subset.

Contribution

It extends Erdos's theorem to algebraic hypergraphs, establishing the existence of large independent subsets under new conditions.

Findings

Generalization of Erdos's theorem to algebraic hypergraphs

Infinite subsets contain large independent subsets if no infinite complete subset exists

Applicable to a broad class of algebraic hypergraphs

Abstract

A theorem of Erdos asserts that every infinite subset of Euclidean n-space R^n has a subset of the same cardinality having no repeated distances. This theorem is generalized here as follows: If (R^n,E) is an algebraic hypergraph that does not have an infinite, complete subset, then every infinite subset of it has an independent subset of the same cardinality.