On the complexity of finding and counting solution-free sets of integers

Meeks, K. and Treglown, A. (2018) On the complexity of finding and counting solution-free sets of integers. Discrete Applied Mathematics, 243, pp. 219-238. (doi: 10.1016/j.dam.2018.02.008)

Abstract

Given a linear equation L, a set A of integers is L-free if A does not contain any ‘non-trivial’ solutions to L. This notion incorporates many central topics in combinatorial number theory such as sum-free and progression-free sets. In this paper we initiate the study of (parameterised) complexity questions involving L-free sets of integers. The main questions we consider involve deciding whether a finite set of integers A has an L-free subset of a given size, and counting all such L-free subsets. We also raise a number of open problems.

• Item Type: Articles
• Additional Information: The first author is supported by a Personal Research Fellowship from the Royal Society of Edinburgh, funded by the Scottish Government, and the second author is supported by EPSRC grant EP/M016641/1.
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Meeks, Dr Kitty
• Authors: Meeks, K., and Treglown, A.
• College/School: College of Science and Engineering > School of Computing Science
• Journal Name: Discrete Applied Mathematics
• Publisher: Elsevier
• ISSN: 0166-218X
• ISSN (Online): 1872-6771
• Published Online: 26 March 2018
• Copyright Holders: Copyright © 2018 The Authors
• First Published: First published in Discrete Applied Mathematics 243:219-238
• Publisher Policy: Reproduced under a Creative Commons License