Boundedly finite-to-one functions

2024/07/14 by Hu, Xiao, Shen, Guozhen
#Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO) #Primary 03E10 #Secondary 03E25

paper · doi:10.48550/arxiv.2407.10183

Abstract

A function is boundedly finite-to-one if there is a natural number k such that each point has at most k inverse images. In this paper, we prove in ZF (i.e., the Zermelo--Fraenkel set theory without the axiom of choice) several results concerning this notion, among which are the following: (1) For each infinite set A and natural number n, there is no boundedly finite-to-one function from S(A) to S_{≤ n}(A), where S(A) is the set of all permutations of A and S_{≤ n}(A) is the set of all permutations of A moving at most n points. (2) For each infinite set A, there is no boundedly finite-to-one function from B(A) to fin(A), where B(A) is the set of all partitions of A such that every block is finite and fin(A) is the set of all finite subsets of A.

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