Sum-free Subsets Research Articles

On Sum-Free Subsets of Abelian Groups

In this paper, we discuss some of the key properties of sum-free subsets of abelian groups. Our discussion has been designed with a broader readership in mind and is hence not overly technical. We consider answers to questions like the following: How many sum-free subsets are there in a given abelian group G? What are its sum-free subsets of maximum cardinality? What is the maximum cardinality of these sum-free subsets? What does a typical sum-free subset of G look like?

On Maximal Sum-Free Sets in Abelian Groups

Balogh, Liu, Sharifzadeh and Treglown [Journal of the European Mathematical Society, 2018] recently gave a sharp count on the number of maximal sum-free subsets of $\{1, \dots, n\}$, thereby answering a question of Cameron and Erdős. In contrast, not as much is know about the analogous problem for finite abelian groups. In this paper we give the first sharp results in this direction, determining asymptotically the number of maximal sum-free sets in both the binary and ternary spaces $\mathbb Z^k_2$ and $\mathbb Z^k_3$. We also make progress on a conjecture of Balogh, Liu, Sharifzadeh and Treglown concerning a general lower bound on the number of maximal sum-free sets in abelian groups of a fixed order. Indeed, we verify the conjecture for all finite abelian groups with a cyclic component of size at least 3084. Other related results and open problems are also presented.

The largest $(k,\ell )$-sum-free subsets

Let $\mathscr{M}_{(2,1)}(N)$ be the infimum of the largest sum-free subset of any set of $N$ positive integers. An old conjecture in additive combinatorics asserts that there is a constant $c=c(2,1)$ and a function $\omega(N)\to\infty$ as $N\to\infty$, such that $cN+\omega(N)<\mathscr{M}_{(2,1)}(N)<(c+o(1))N$. The constant $c(2,1)$ is determined by Eberhard, Green, and Manners, while the existence of $\omega(N)$ is still wide open. In this paper, we study the analogous conjecture on $(k,\ell)$-sum-free sets and restricted $(k,\ell)$-sum-free sets. We determine the constant $c(k,\ell)$ for every $(k,\ell)$-sum-free sets, and confirm the conjecture for infinitely many $(k,\ell)$.

L-functions and sum-free sets

For set \(A\subset \mathbb{F}_p^*\) define by \(\mathsf{sf}(A)\) the size of the largest sum-free subset of A. Alon and Kleitman [3] showed that \(\mathsf{sf} (A) \ge |A|/3+O(|A|/p)\). We prove that if \(\mathsf{sf}(A)-|A|/3\) is small then the set A must be uniformly distributed on cosets of each large multiplicative subgroup. Our argument relies on irregularity of distribution of multiplicative subgroups on certain intervals in \(\mathbb{F}p\).

Sum-Free Sets of Integers with a Forbidden Sum

A set of integers is sum-free if it contains no solution to the equation $x+y=z$. We study sum-free subsets of the set of integers $[n]=\{ 1, \ldots ,n \}$ for which the integer $2n+1$ cannot be re...

THE MAXIMUM SIZE OF -SUM-FREE SETS IN CYCLIC GROUPS

A subset$A$of a finite abelian group$G$is called$(k,l)$-sum-free if the sum of$k$(not necessarily distinct) elements of$A$never equals the sum of$l$(not necessarily distinct) elements of $A$. We find an explicit formula for the maximum size of a$(k,l)$-sum-free subset in$G$for all$k$and$l$in the case when$G$is cyclic by proving that it suffices to consider$(k,l)$-sum-free intervals in subgroups of $G$. This simplifies and extends earlier results by Hamidoune and Plagne [‘A new critical pair theorem applied to sum-free sets in abelian groups’,Comment. Math. Helv. 79(1) (2004), 183–207] and Bajnok [‘On the maximum size of a$(k,l)$-sum-free subset of an abelian group’,Int. J. Number Theory 5(6) (2009), 953–971].

On the structure of large sum-free sets of integers

A set of integers is called sum-free if it contains no triple $(x,y,z)$ of not necessarily distinct elements with $x+y=z$. In this paper, we provide a structural characterisation of sum-free subsets of $\{1,2,\ldots,n\}$ of density at least $2/5-c$, where $c$ is an absolute positive constant. As an application, we derive a stability version of Hu's Theorem [Proc. Amer. Math. Soc. 80 (1980), 711-712] about the maximum size of a union of two sum-free sets in $\{1,2,\ldots,n\}$. We then use this result to show that the number of subsets of $\{1,2,\ldots,n\}$ which can be partitioned into two sum-free sets is $\Theta(2^{4n/5})$, confirming a conjecture of Hancock, Staden and Treglown [arXiv:1701.04754].

Symmetric complete sum-free sets in cyclic groups

We present constructions of symmetric complete sum-free sets in general finite cyclic groups. It is shown that the relative sizes of the sets are dense in [0, 1/3], answering a question of Cameron, and that the number of those contained in the cyclic group of order n is exponential in n. For primes p, we provide a full characterization of the symmetric complete sum-free subsets of ℤp of size at least (1/3−c)·p, where c > 0 is a universal constant.

The growth rate of tri-colored sum-free sets

The growth rate of tri-colored sum-free sets, Discrete Analysis 2018:12, 10 pp. This paper contributes to the remarkable collection of results that followed in the wake of the 2016 breakthrough by Ellenberg and Gijswijt on the cap set problem, which asks for the maximal size of a 3-term progression-free subset of $\mathbb{F}_3^n$. The polynomial method of Ellenberg and Gijswijt, who followed the lead of Croot, Lev, and Pach after whom the method is now named, showed for the first time that the size of such a set is bounded by a polynomial in the size of the ambient space. More specifically, they showed that a cap set in $\mathbb{F}_3^n$ can be of size at most $(2.756)^n$. This paper considers a variant of the cap set problem, namely the question of how large a tri-coloured sum-free subset of $\mathbb{F}_3^n$ can be. By a tri-coloured sum-free subset we mean a collection of triples $(a_i,b_i,c_i)$ in $(\mathbb{F}_3^n)^3$ such that $a_i+b_j+c_k=0$ if and only if $i=j=k$. Note that a cap set $A\subseteq \mathbb{F}_3^n$ gives rise to a tri-coloured sum-free set, namely the collection of triples $\{(a,a,a): a\in A\}$. It is therefore not entirely surprising that the Croot-Lev-Pach polynomial method can be used to show that if $q$ is a prime power, then a tri-coloured sum-free set in $\mathbb{F}_q^n$ can have size at most $3\theta^n$, where $\theta$ is the solution to an explicit optimisation problem. This is one of the results appearing in the paper ["On cap sets and the group-theoretic approach to matrix multiplication"](http://discreteanalysisjournal.com/article/1245), by Blasiak, Church, Cohn, Grochow, Naslund, Sawin, and Umans, which Discrete Analysis published earlier this year. In the present paper the authors show this bound to be tight to within a subexponential factor. The lower bound is based on a construction of an $S_3$-symmetric probability distribution on $\{(a,b,c)\in\mathbb{Z}_{\geq 0}^3:a+b+c=n\}$ such that its marginal achieves the maximum entropy among all probability distributions on $\{0,1,…,n\}$ with mean $n/3$. In answer to a conjecture which was posed in the original preprint version of this article, the existence of such a distribution was established by Pebody, whose [article](http://discreteanalysisjournal.com/article/3733-proof-of-a-conjecture-of-kleinberg-sawin-speyer) is being published in Discrete Analysis alongside the present paper. The question of whether the Croot-Lev-Pach polynomial method also yields a tight bound for the cap set problem remains open.

Three questions of Bertram on locally maximal sum-free sets

Let G be a finite group, and S a sum-free subset of G. The set S is locally maximal in G if S is not properly contained in any other sum-free set in G. If S is a locally maximal sum-free set in a finite abelian group G, then $$G=S\cup SS\cup SS^{-1}\cup \sqrt{S}$$ , where $$SS=\{xy|~x,y\in S\}$$ , $$SS^{-1}=\{xy^{-1}|~x,y\in S\}$$ and $$\sqrt{S}=\{x\in G|~x^2\in S\}$$ . Each set S in a finite group of odd order satisfies $$|\sqrt{S}|=|S|$$ . No such result is known for finite abelian groups of even order in general. In view to understanding locally maximal sum-free sets, Bertram asked the following questions: In this paper, we answer question (i) in the negative, then (ii) and (iii) in affirmative.

Sharp bound on the number of maximal sum-free subsets of integers

Cameron and Erdős [6] asked whether the number of maximal sum-free sets in \{1, \dots , n\} is much smaller than the number of sum-free sets. In the same paper they gave a lower bound of 2^{\lfloor n/4 \rfloor } for the number of maximal sum-free sets. Here, we prove the following: For each 1\leq i \leq 4 , there is a constant C_i such that, given any n\equiv i \mod 4 , \{1, \dots , n\} contains (C_i+o(1)) 2^{n/4} maximal sum-free sets. Our proof makes use of container and removal lemmas of Green [11, 12], a structural result of Deshouillers, Freiman, Sós and Temkin [7] and a recent bound on the number of subsets of integers with small sumset by Green and Morris [13]. We also discuss related results and open problems on the number of maximal sum-free subsets of abelian groups.

Symmetric Complete Sum-free Sets in Cyclic Groups

We present constructions of symmetric complete sum-free sets in general finite cyclic groups. It is shown that the relative sizes of the sets are dense in [0,13], answering a question of Cameron, and that the number of those contained in the cyclic group of order n is exponential in n. For primes p, we provide a full characterization of the symmetric complete sum-free subsets of Zp of size at least (13−c)⋅p, where c>0 is a universal constant.

Sum-free subsets of finite abelian groups of type III

A finite abelian group G of cardinality n is said to be of type III if every prime divisor of n is congruent to 1 modulo 3. We obtain a classification theorem for sum-free subsets of largest possible cardinality in a finite abelian group G of type III. This theorem, when taken together with known results, gives a complete characterisation of sum-free subsets of the largest cardinality in any finite abelian group G. We supplement this result with a theorem on the structure of sum-free subsets of cardinality “close” to the largest possible in a type III abelian group G. We then give two applications of these results. Our first application allows us to write down a formula for the number of orbits under the natural action of Aut(G) on the set of sum-free subsets of G of the largest cardinality when G is of the form (Z/mZ)r, with all prime divisors of m congruent to 1 modulo 3, thereby extending a result of Rhemtulla and Street. Our second application provides an upper bound for the number of sum-free subsets of G. For finite abelian groups G of type III and with a given exponent this bound is substantially better than that implied by the bound for the number of sum-free subsets in an arbitrary finite abelian group, due to Green and Ruzsa.

The number of additive triples in subsets of abelian groups

AbstractA set of elements of a finite abelian group is called sum-free if it contains no Schur triple, i.e., no triple of elementsx,y,zwithx+y=z. The study of how large the largest sum-free subset of a given abelian group is had started more than thirty years before it was finally resolved by Green and Ruzsa a decade ago. We address the following more general question. Suppose that a setAof elements of an abelian groupGhas cardinalitya. How many Schur triples mustAcontain? Moreover, which sets ofaelements ofGhave the smallest number of Schur triples? In this paper, we answer these questions for various groupsGand ranges ofa.

Counting independent sets in graphs

In this short survey article, we present an elementary, yet quite powerful, method of enumerating independent sets in graphs. This method was first employed more than three decades ago by Kleitman and Winston and has subsequently been used numerous times by many researchers in various contexts. Our presentation of the method is illustrated with several applications of it to ‘real-life’ combinatorial problems. In particular, we derive bounds on the number of independent sets in regular graphs, sum-free subsets of {1,…,n}, and C4-free graphs and give a short proof of an analogue of Roth’s theorem on 3-term arithmetic progressions in sparse random sets of integers which was originally formulated and proved by Kohayakawa, Łuczak, and Rödl.

The Sharp Threshold for Maximum-Size Sum-Free Subsets in Even-Order Abelian Groups

We study sum-free sets in sparse random subsets of even-order abelian groups. In particular, we determine the sharp threshold for the following property: the largest such set is contained in some maximum-size sum-free subset of the group. This theorem extends recent work of Balogh, Morris and Samotij, who resolved the caseG= ℤ2n, and who obtained a weaker threshold (up to a constant factor) in general.

Følner sequences and sum-free sets

Erd\H{o}s showed that every set of $n$ positive integers contains a subset of size at least $n/(k+1)$ containing no solutions to $x_1 + \cdots + x_k = y$. We prove that the constant $1/(k+1)$ here is best possible by showing that if $(F_m)$ is a multiplicative F{\o}lner sequence in $\mathbf{N}$ then $F_m$ has no $k$-sum-free subset of size greater than $(1/(k+1)+o(1))|F_m|$. This provides a new proof and a generalisation of a recent theorem of Eberhard, Green, and Manners.

Sets of integers with no large sum-free subset

Answering a question of P. Erdos from 1965, we show that for every eps>0 there is a set A of n integers with the following property: every subset A' of A with at least (1/3 + eps)n elements contains three distinct elements x,y,z with x + y = z.

Finding Linear Patterns of Complexity One

We study the following generalization of Roth's theorem for 3-term arithmetic progressions. For s>1, define a nontrivial s-configuration to be a set of s(s+1)/2 integers consisting of s distinct integers x_1,...,x_s as well as all the averages (x_i+x_j)/2. Our main result states that if a set A contained in {1,2,...,N} has density at least (log N)^{-c(s)} for some positive constant c(s)>0 depending on s, then A contains a nontrivial s-configuration. We also deduce, as a corollary, an improvement of a problem involving sumfree subsets.

Random sum-free subsets of abelian groups

We characterize the structure of maximum-size sum-free subsets of a random subset of an abelian group G. In particular, we determine the threshold above which, with high probability as |G| → ∞, each such subset is contained in some maximum-size sum-free subset of G, whenever q divides |G| for some (fixed) prime q with q ≡ 2 (mod 3). Moreover, in the special case G = ℤ2n , we determine the sharp threshold for the above property. The proof uses recent ‘transference’ theorems of Conlon and Gowers, together with stability theorems for sum-free sets of abelian groups.