Let G be a finite Abelian group of order |G| = n, and let S = g1 · . . . · gn−1 be a sequence over G such that all nonempty zero-sum subsequences of S have the same length. In this paper, we completely determine the structure of these sequences.

In 1963, Graham [1] proved that all integers greater than 77 (but not 77 itself) can be partitioned into distinct positive integers whose reciprocals sum to 1. He further conjectured [2, Section D11] that for any sufficiently large integer, it can be partitioned into squares of distinct positive integers whose reciprocals sum to 1. In this study, we establish the exact bound for existence of su...

The purpose of this note is to construct a non-hopfian and finitely generated group 5 which is in no way complicated. This group S is a generalised free square of the free nilpotent group A of class two on two generators. Since A satisfies the maximum condition, S differs radically from the non-hopfian groups constructed by Graham Higman [ l ] , with which it may be compared. I t may perhaps be...

We show that asymptotically almost surely a tree with m edges decomposes the complete bipartite graph K 2m,2m , a result connected to a conjecture of Graham and Häggkvist. The result also implies that asymptotically almost surely a tree with m edges decomposes the complete graph with O(m 2) edges. An ingredient of the proof consists in showing that the bipartition classes of the base tree of a ...

The induced restricted versions of the vector space Ramsey theorem and of the Graham-Rothschild parameter set theorem are proved.

In this note, we sharpen work of Ulas to provide what is, in some sense, the minimal counterexample to a “conjecture” of Erdős and Graham about square values of products of disjoint blocks of consecutive integers.

Let n be a positive integer and let S be a sequence of n integers in the interval [0, n − 1]. If there is an r such that any nonempty subsequence with sum ≡ 0 (mod n) has length = r, then S has at most two distinct values. This proves a conjecture of R. L. Graham. A previous result of P. Erdős and E. Szemerédi shows the validity of this conjecture if n is a large prime number.

We construct non-periodic 2-colorings that avoid long monochromatic progressions having odd common differences. Also we prove that the set of all arithmetic progressions with common differences in (N!−1) ∪ N! ∪ (N!+1)− {0} does not have the 2-Ramsey property.

In this paper we relate the Fefferman–Graham ambient metric construction for conformal manifolds to the approach to conformal geometry via the canonical Cartan connection. We show that from any ambient metric that satisfies a weakening of the usual normalisation condition, one can construct the conformal standard tractor bundle and the normal standard tractor connection, which are equivalent to...

It is conjectured by Erdős, Graham and Spencer that if 1 ≤ a1 ≤ a2 ≤ · · · ≤ as with ∑s i=1 1/ai < n − 1/30, then this sum can be decomposed into n parts so that all partial sums are ≤ 1. In this note we propose a counterexample which gives a negative answer to this conjecture.