Csaba Sándor

An upper bound for Hilbert cubes

In this note we give a new upper bound for the largest size of subset of {1,2,...,n} not containing a k-cube.

Groups, partitions and representation functions

Let X be a semigroup written additively and h ≥ 2 a fixed integer. Let x be an element of X and A1, . . . , Ah be nonempty subsets of X. Let RA1+···+Ah(x) denote the number of solutions of the equation a1 + · · · + ah = x, where ai ∈ Ai. In this paper for X = N we give a necessary and sufficient condition such that the equality RA1+A2(n) = RX\A1+X\A2(n) holds from a certain point on. We study similar questions when X = Zm and in general when X = G, where G is a finite additive group.

On the maximum values of the additive representation functions

Let A and B be infinite sequences of nonnegative integers. For a positive integer n, let RA(n) denote the number of representations of n as the sum of two terms from A. Let sA(x) denote the maximum value of RA(n) up to x and dA,B(x) denote the distance of the sequences A and B. In this paper, we study the connection between sA(x), sB(x) and dA,B(x). We improve a result of Haddad and Helou about the Erdős–Turan conjecture.

On generalized Stanley sequences

Let

On infinite multiplicative Sidon sets

{\mathbb{N}}

k

N denote the set of all nonnegative integers. Let

Multiplicative bases and an Erdős problem

{k \ge 3}