must hold, where A(x) denotes the number of elements of A up to x. It is quite natural to ask how much the situation changes if we cut A into two parts, A' and A ", and demand only that no a i' a, should coincide with any a i ' -a,' . This question was proposed by Erdős and Graham in [2], and it seemed likely that no considerable increase can be achieved in the density of A . We shall show, how...

Abstract. Let A = {as(mod ns)}ks=1 and B = {bt(mod mt)} l t=1 be two systems of residue classes. If |{1 6 s 6 k : x ≡ as (mod ns)}| and |{1 6 t 6 l : x ≡ bt (mod mt)}| are equal for all x ∈ Z, then A and B are said to be covering equivalent. In this paper we characterize the covering equivalence in a simple and new way. Using the characterization we partially confirm a conjecture of R. L. Graha...

On pseudo-Riemannian manifolds of even dimension n ≥ 4, with everywhere vanishing (Fefferman-Graham) obstruction tensor, we construct a complex of conformally invariant differential operators. The complex controls the infinitesimal deformations of obstruction-flat structures, and, in the case of Riemannian signature the complex is elliptic.

For positive integers k and n the group of perfect k-shuffles with a deck of kn cards is a subgroup of the symmetric group Skn. The structure of these groups was found for k = 2 by Diaconis, Graham, and Kantor and for k ≥ 3 and a deck of km cards by Medvedoff and Morrison. They also conjectured that for k = 4 and deck size 2m, m odd, the group is isomorphic to the group of affine transformation...

The altitude of a graphG, denoted f(G), is the largest integer k such that under each ordering of E(G), there exists a path of length k which traverses edges in increasing order. In 1971, Chvátal and Komlós asked for f(Kn), where Kn is the complete graph on n vertices. In 1973, Graham and Kleitman proved that f(Kn) ≥ √ n− 3/4− 1/2 and in 1984, Calderbank, Chung, and Sturtevant proved that f(Kn)...

For a tree T , the graph X is T -decomposable if there exists a partition of the edge set of X into isomorphic copies of T . In 1963, Ringel conjectured that K2m+1 can be decomposed by any tree with m edges. Graham and Häggkvist conjectured more generally that every 2m-regular graph can be decomposed by any tree with m edges. Fink showed in 1994 that for any directed tree T with m arcs, the dir...

In this paper, we construct, given an integer r ≥ 5, an infinite family of r non-overlapping blocks of five consecutive integers with the property that their product is always a perfect square. In this particular situation, this answers a question of Erdős and Graham in the negative.

A brief introduction to the theory of ordered sets and lattice theory is given. To illustrate proof techniques in the theory of ordered sets, a generalization of a conjecture of Daykin and Daykin, concerning the structure of posets that can be partitioned into chains in a “strong” way, is proved. The result is motivated by a conjecture of Graham’s concerning probability correlation inequalities...

We consider the problem of packing n disks of unit diameter in the plane so as to minimize the second moment about their centroid. Our main result is an algorithm which constructs packings that are optimal among hexagonal packings. Using the algorithm, we prove that, except for n = 212, the n-point packings obtained by Graham and Sloane [1] are optimal among hexagonal packings. We also prove a ...

This note describes a polynomial space proof of the Graham–Pollak theorem.