Let γ(k, p) denote Waring’s number (mod p) and δ(k, p) denote the ± Waring’s number (mod p). We use sum-product estimates for |nA| and |nA− nA|, following the method of Glibichuk and Konyagin, to estimate γ(k, p) and δ(k, p). In particular, we obtain explicit numerical constants in the Heilbronn upper bounds: γ(k, p) ≤ 83 k1/2, δ(k, p) ≤ 20 k1/2 for any positive k not divisible by (p− 1)/2.

Let G be a connected finite graph. The average distance μ(G) of G is the average of the distances between all pairs of vertices of G. For a positive integer k a k-packing of G is a subset S of the vertex set of G such that the distance between any two vertices in S is greater than k. The k-packing number βk(G) of G is the maximum cardinality of a k-packing of G. We prove upper bounds on the ave...

Let S be a set of horizontal line segments, or bars, in the plane. We say that G is a bar visibility graph, and S its bar visibility representation, if there exists a one-to-one correspondence between vertices of G and bars in S, such that there is an edge between two vertices in G if and only if there exists an unobstructed vertical line of sight between their corresponding bars. If bars are a...

The function h(k) represents the smallest number m such that every sequence of m consecutive integers contains an integer coprime to the first k primes. We give a new computational method for calculating strong upper bounds on h(k).

For an absolutely continuous probability measure on Rd and a nonnegative integer k, let ~ sk( ;0) denote the probability that the convex hull of k + d + 1 random points which are i.i.d. according to contains the origin 0. For d and k given, we determine a tight upper bound on ~ sk( ;0), and we characterize the measures in Rd which attain this bound. As we will see, this result can be considered...

The function h(k) represents the smallest number m such that every sequence of m consecutive integers contains an integer coprime to the first k primes. We give a new computational method for calculating strong upper bounds on h(k).

In this paper, we provide a new upper bound for the α-domination number. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another well-known upper bound for the classical domination in graphs. We also prove similar upper bounds for the α-rate domination number, which combines the concepts of...

A new upper bound is given for the cycle-complete graph Ramsey number r(C,,,, K„), the smallest order for a graph which forces it to contain either a cycle of order m or a set of ri independent vertices . Then, another cycle-complete graph Ramsey number is studied, namely r( :C,,,, K„) the smallest order for a graph which forces it to contain either a cycle of order 1 for some I satisfying 3 :1...