The covering number of the difference sets in partitions of $G$-spaces and groups

TL;DR

This paper establishes bounds on the covering numbers of difference sets in finite partitions of groups and G-spaces, providing partial solutions to a problem posed by Protasov in 1995.

Contribution

It proves new bounds on covering numbers of difference sets in partitions of groups and G-spaces, advancing understanding of their combinatorial structure.

Findings

Either all cells have covering number at most n or some cell has covering number less than n.

The results apply to partitions of groups and G-spaces.

Provides partial answers to Protasov's 1995 problem.

Abstract

We prove that for every finite partition $G=A_1\cup\dots\cup A_n$ of a group $G$ either $cov(A_iA_i^{-1})\le n$ for all cells $A_i$ or else $cov(A_iA_i^{-1}A_i)<n$ for some cell $A_i$ of the partition. Here $cov(A)=\min\{|F|:F\subset G,\;G=FA\}$ is the covering number of $A$ in $G$. A similar result is proved also of partitions of $G$-spaces. This gives two partial answers to a problem of Protasov posed in 1995.