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

Let $G=C_n\oplus C_{mn}$ with $n\geq 2$ and $m\geq 1$, and let $k\in [0,n-1]$. It is known that any sequence of $mn+n-1+k$ terms from $G$ must contain a nontrivial zero-sum of length at most $mn+n-1-k$. The associated inverse question is to…

In the area of forbidden subposet problems we look for the largest possible size $La(n,P)$ of a family $\mathcal{F}\subseteq 2^{[n]}$ that does not contain a forbidden inclusion pattern described by $P$. The main conjecture of the area…

Let $G = (G,+)$ be an additive group. The sumset theory of Pl\"unnecke and Ruzsa gives several relations between the size of sumsets $A+B$ of finite sets $A, B$, and related objects such as iterated sumsets $kA$ and difference sets $A-B$,…

Frankl's union-closed sets conjecture states that in every finite union-closed set of sets, there is an element that is contained in at least half of the member-sets (provided there are at least two members). The conjecture has an…

The inducibility of a graph $H$ measures the maximum number of induced copies of $H$ a large graph $G$ can have. Generalizing this notion, we study how many induced subgraphs of fixed order $k$ and size $\ell$ a large graph $G$ on $n$…

A subset $A$ of the integers is a $B_k[g]$ set if the number of multisets from $A$ that sum to any fixed integer is at most $g$. Let $F_{k,g}(n)$ denote the maximum size of a $B_k[g]$ set in $\{1,\dots, n\}$. In this paper we improve the…

Given a set of $n$ real numbers, if the sum of elements of every subset of size larger than $k$ is negative, what is the maximum number of subsets of nonnegative sum? In this note we show that the answer is $\binom{n-1}{k-1} +…

Let $G$ be an additive finite abelian group with exponent $\exp(G)$. For $L\subseteq \mathbb N$, let $\mathsf{s}_{L}(G)$ be the smallest integer $\ell$ such that every sequence $S$ over $G$ of length $\ell$ has a zero-sum subsequence $T$ of…

Let $e(n,s)$ denote the maximum size of a family $\mathcal{F}$ of subsets of an $n$-element set that contains no $s$ pairwise disjoint members. In 1968, answering a question of Erd\H{o}s, Kleitman determined $e(sm-1,s)$ and $e(sm,s)$ for…

We study the number of $s$-element subsets $J$ of a given abelian group $G$, such that $|J+J|\leq K|J|$. Proving a conjecture of Alon, Balogh, Morris and Samotij, and improving a result of Green and Morris, who proved the conjecture for $K$…

For integers $n\ge s\ge 2$ let $e(n,s)$ denote the maximum of $|\mathcal F|,$ where $\mathcal F$ is a family of subsets of an $n$-element set and $\mathcal F$ contains no $s$ pairwise disjoint members. Half a century ago, solving a…

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 $p_c \approx \sqrt{\log n / n}$ above which, with high probability as $|G| \to \infty$,…

Three types of Cantor sets are studied.For any integer $m\ge 4$, we show that every real number in $[0,k]$ is the sum of at most $k$ $m$-th powers of elements in the Cantor ternary set $C$ for some positive integer $k$, and the smallest…

A family of sets F is said to be union-closed if A \cup B is in F for every A and B in F. Frankl's conjecture states that given any finite union-closed family of sets, not all empty, there exists an element contained in at least half of the…

We determine, within 1, the value of N for which sum (s1 choose i)(s2 choose N)(s1 choose N-i)(N choose i) achieves its maximum value. Here s1 and s2 are fixed integers. This problem arises in studying the most likely value for the size of…

We consider the Sobolev (Bessel potential) spaces H^ell(R^d, C), and their standard norms || ||_ell (with ell integer or noninteger). We are interested in the unknown sharp constant K_{ell m n d} in the inequality || f g ||_{ell} \leqs…

The $n$-linear Bohnenblust-Hille inequality asserts that there is a constant $C_{n}\in\lbrack1,\infty)$ such that the $\ell_{\frac{2n}{n+1}}$-norm of $(U(e_{i_{^{1}}},...,e_{i_{n}}))_{i_{1},...i_{n}=1}^{N}$is bounded above by $C_{n}$ times…

The main result of this paper is the following: for all $b \in \mathbb Z$ there exists $k=k(b)$ such that \[ \max \{ |A^{(k)}|, |(A+u)^{(k)}| \} \geq |A|^b, \] for any finite $A \subset \mathbb Q$ and any non-zero $u \in \mathbb Q$. Here,…

For a constant $\gamma \in[0,1]$ and a graph $G$, let $\omega_{\gamma}(G)$ be the largest integer $k$ for which there exists a $k$-vertex subgraph of $G$ with at least $\gamma\binom{k}{2}$ edges. We show that if $0<p<\gamma<1$ then…

In 2008, Cusick {\it et al.} conjectured that certain elementary symmetric Boolean functions of the form $\sigma_{2^{t+1}l-1, 2^t}$ are the only nonlinear balanced ones, where $t$, $l$ are any positive integers, and…