Erd\H{o}s' problem and $(n, \frac{1}{3})$-separated sets

TL;DR

This paper introduces a dynamical version of Erdős' problem in Ramsey theory, focusing on circle maps, and provides a positive solution for this new formulation.

Contribution

It formulates a dynamical analogue of Erdős' problem and demonstrates a positive answer specifically for circle maps, extending classical combinatorial results.

Findings

Established a dynamical version of Erdős' problem

Proved positive results for circle maps in this context

Bridged combinatorial and dynamical systems theory

Abstract

Inspired by the Erd\H{o}s' problem in Ramsey theory, we propose a dynamical version of the problem and answer it positively for circle maps.