We show that certain natural aperiodic sum-free sets are incomplete that is that there are infinitely many n not in S which are not a sum of two element of S.

We present new explicit lower bounds for some Ramsey numbers. All the graphs are cyclic, and are on a prime number of vertices. We give a partial probabilistic analysis which suggests that the cyclic Ramsey numbers grow ...

Cameron has introduced a natural bijection between the set of one way in nite binary sequences and the set of sum-free sets (of positive integers), and observed that a sum-free set is ultimately periodic only if the ...

We determine lower bounds for the number of random binary vectors, chosen uniformly from vectors of weight k, needed to obtain a dependent set.

We show that if S_1 is a strongly complete sum-free set of positive integers, and if S_0 is a finite sum-free set, then with positive probability a random sum-free set U contains S_0 and is contained in S_0\cup S_1. As ...

We show that for every k greater or equal than 3 the number of subsets of {1,2,...,n} containing no solution to x_1 + x_2 + ... + x_k = y, where the x_i need not be distinct, is at most c2^{\alpha n}, where \alpha = (k-1)/k.

In this note we shall prove the following curious identity of sums of powers of the partial sum of binomial coefficients.