Oscilation stability for continuous monotone surjections

TL;DR

This paper proves a stability result for colorings of nondecreasing surjections on infinite sequences, showing that a finite number of colors can be approximated by a cube within a specified epsilon margin.

Contribution

It establishes a new combinatorial stability theorem for continuous monotone surjections on infinite sequences, extending previous results in the area.

Findings

Existence of a finite number t for colorings with stability properties

Colorings contain a cube within epsilon-fattening of t colors

Applicable to nondecreasing surjections on b-ary sequences

Abstract

We prove that for every integer $b\geqslant 2$ and positive real $\varepsilon$ there exists a finite number $t$ such that for every finite coloring of the nondecreasing surjections from $b^\omega$ onto $b^\omega$, there exist $t$ many colors such that their $\varepsilon$-fattening contains a cube.