Finite versus infinite: an insufficient shift

TL;DR

This paper investigates the properties of the shift graph on infinite subsets of natural numbers, showing it is not minimal among a broad class of graphs defined via Borel functions on Polish spaces, thus answering a key open question.

Contribution

It demonstrates that the shift graph is not minimal among graphs defined by Borel functions on Polish spaces, resolving a major open problem.

Findings

Shift graph is not minimal among graphs of the form G_f.

Answers the primary open question from Kechris et al. 1999.

Provides insights into the structure of graphs on infinite subsets.

Abstract

The shift graph is defined on the space of infinite subsets of natural numbers by letting two sets be adjacent if one can be obtained from the other by removing its least element. We show that this graph is not a minimum among the graphs of the form $G_{f}$ defined on some Polish space $X$, where two distinct points are adjacent if one can be obtained from the other by a given Borel function $f:X\to X$. This answers the primary outstanding question from \cite{Kechris19991}.