Related papers: On solution-free sets of integers

In this paper, we are interested in a generalization of the notion of sum-free sets. We address a conjecture first made in the 90s by Chung and Goldwasser. Recently, after some computer checks, this conjecture was formulated again by…

For a rational number $r>1$, a set $A$ of positive integers is called an $r$-multiple-free set if $A$ does not contain any solution of the equation $rx = y$. The extremal problem on estimating the maximum possible size of $r$-multiple-free…

Cameron and Erd\H{o}s raised the question of how many maximal sum-free sets there are in $\{1, \dots , n\}$, giving a lower bound of $2^{\lfloor n/4 \rfloor }$. In this paper we prove that there are in fact at most $2^{(1/4+o(1))n}$ maximal…

In this paper we study sum-free subsets of the set $\{1,...,n\}$, that is, subsets of the first $n$ positive integers which contain no solution to the equation $x + y = z$. Cameron and Erd\H{o}s conjectured in 1990 that the number of such…

A subset $A$ of a given finite abelian group $G$ is called $(k,l)$-sum-free if the sum of $k$ (not necessarily distinct) elements of $A$ does not equal the sum of $l$ (not necessarily distinct) elements of $A$. We are interested in finding…

A 3-simplex is a collection of four sets A_1,...,A_4 with empty intersection such that any three of them have nonempty intersection. We show that the maximum size of a set system on n elements without a 3-simplex is $2^{n-1} +…

Let A be a subset of an abelian group G. We say that A is sum-free if there do not exist x,y and z in A satisfying x + y = z. We determine, for any G, the cardinality of the largest sum-free subset of G. This equals c(G)|G| where c(G) is a…

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…

Given a linear equation of the form $a_1x_1 + a_2x_2 + a_3x_3 = 0$ with integer coefficients $a_i$, we are interested in maximising the number of solutions to this equation in a set $S \subseteq \mathbb{Z}$, for sets $S$ of a given size. We…

Let $k$ and $n$ be fixed positive integers. For each prime power $q\geqslant k\geqslant 3$, we show that any subset $A\subseteq \mathbb{Z}_q^n$ free of $k$-term arithmetic progressions has size $|A|\leqslant c_k(q)^n$ with a constant…

Fix $A$, a family of subsets of natural numbers, and let $G_A(n)$ be the maximum cardinality of a subset of $\{1,2,..., n\}$ that does not have any subset in $A$. We consider the general problem of giving upper bounds on $G_A(n)$ and give…

We count the ordered sum-free triplets of subsets in the group $\mathbb{Z}/p\mathbb{Z}$, i.e., the triplets $(A,B,C)$ of sets $A,B,C \subset \mathbb{Z}/p\mathbb{Z}$ for which the equation $a+b=c$ has no solution with $a\in A$, $b \in B$ and…

Let $n$ and $k$ be integers. A set $A\subset\mathbb{Z}/n\mathbb{Z}$ is $k$-free if for all $x$ in $A$, $kx\notin A$. We determine the maximal cardinality of such a set when $k$ and $n$ are coprime. We also study several particular cases and…

We expand the class of linear symmetric equations for which large sets with no non-trivial solutions are known. Our idea is based on first finding a small set with no solutions and then enlarging it to arbitrary size using a…

We study subsets of $\mathbb{F}_p^n$ that do not contain progressions of length $k$. We denote by $r_k(\mathbb{F}_p^n)$ the cardinality of such subsets containing a maximal number of elements. In this paper we focus on the case $k=p$ and…

The paper considers the problem of finding the largest possible set P(n), a subset of the set N of the natural numbers, with the property that a number is in P(n) if and only if it is a sum of n distinct naturals all in P(n) or none in…

For a given subset $A\subseteq \mathbb F_q^*$, we study the problem of finding a large packing set $B$ of $A$, that is, a set $B \subseteq \mathbb F_q^*$ such that $|AB|=|A||B|$. We prove the existence of such a $B$ of size $|B|\ge…

A finite set $A$ of integers is square-sum-free if there is no subset of $A$ sums up to a square. In 1986, Erd\H os posed the problem of determining the largest cardinality of a square-sum-free subset of $\{1, ..., n \}$. Answering this…

An old problem of Moser asks: how large of a union-free subfamily does every family of m sets have? A family of sets is called union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to…

Cameron and Erd\H{o}s asked whether the number of \emph{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…