A Note on Combinatorial Derivation

2012/10/29 by Joshua Erde, Erde, Joshua
Computer Science · Mathematics · #Advanced Graph Theory Research #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #General Topology (math.GN) #Limits and Structures in Graph Theory #math.CO #math.GN

paper · pdf · doi:10.48550/arxiv.1210.7622

5 pages, fixed error in statement of Theorem 1

openalex publication_date 2012/10/29 · arxiv created 2014/09/29 · arxiv updated 2014/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given an infinite group G and a subset A of G we let Δ(A) = g ∈ G : |gA ∩ A| =∞ (this is sometimes called the combinatorial derivation of A). A subset A of G is called large if there exists a finite subset F of G such that FA=G. We show that given a large set X, and a decomposition X=A₁ ∪ ... ∪ A_n, there must exist an i such that Δ(A_i) is large. This answers a question of Protasov. We also answer a number of related questions of Protasov.

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