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.

Highlights

  • Sets of integers which do not contain any solutions to some linear equation have received a lot of attention in the field of combinatorial number theory

  • We are primarily concerned with determining the size of the largest subset of an arbitrary set of integers A which avoids solutions to a specified linear equation L; in particular, we focus on sum-free and progression-free sets, but many of our results generalise to larger families of linear equations

  • We have shown that the basic problem of deciding whether a given input set A ⊆ Z contains an L-free subset of size at least k is NP-complete when L is any linear equation of the form a1x1 + · · · + a x = by, the problem is solvable in polynomial time whenever L is a linear equation with only two variables

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Summary

Introduction

Sets of integers which do not contain any solutions to some linear equation have received a lot of attention in the field of combinatorial number theory. We prove that the problem of determining whether a finite set of integers contains a sum-free (or progression-free) subset of a given size is NP-complete (see Theorem 4). We prove that the problem of determining whether a finite set of integers A contains a sum-free subset covering a cth proportion of A is NP-complete for any fixed c > 1/3 (see Theorem 18). It is essentially trivial to see that the problem of determining whether a finite set of integers A contains a sum-free (or progression-free) subset of a given size k is in FPT, when parameterised by k (see Proposition 12). A result of Thurley [43] implies that the analogous problem, for counting the number of sum-free (or progression-free) subsets of A of a given size k, belongs to FPT when parameterised by |A| − k (see Theorem 20). Perhaps surprisingly, it is unlikely that this problem is in FPT when instead we parameterise by k (see Theorem 23)

The decision problem
Approximating the size of the largest L-Free Subset
L-free subsets of arbitrary sets of integers
Parameterised complexity of the decision problem
L-free subsets covering a given fraction of elements
An extension version of the problem
Conclusions and open problems

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