Let ES(n) denote the least integer such that among any ES(n) points in general position in the plane there are always n in convex position. In 1935, P. Erdős and G. Szekeres showed that ES(n) exists and ES(n) ≤ ` 2n−4 n−2 ́ + 1. Six decades later, the upper bound was slightly improved by Chung and Graham, a few months later it was further improved by Kleitman and Pachter, and another few months ...

It is conjectured by Erdős, Graham and Spencer that if 1 ≤ a1 ≤ a2 ≤ · · · ≤ as are integers 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. This is not true for ∑s i=1 1/ai = n − 1/30 as shown by a1 = · · · = an−2 = 1, an−1 = 2, an = an+1 = 3, an+2 = · · · = an+5 = 5. In 1997 Sandor proved that Erdős–Graham–Spencer conjecture is true ...

We define two notions of generation between the various optimal packings Qm of m congruent disks in a subset K of R2. The first one that we call weak generation consists in getting Qn by removing m − n disks from Qm and by displacing the n remaining congruent disks which grow continuously and do not overlap. During a weak generation of Qn from Qm, we consider the contact graphs G(t) of the inte...

We prove an old conjecture of Erdős and Graham on sums of unit fractions: There exists a constant b > 0 such that if we r-color the integers in [2, br ], then there exists a monochromatic set S such that ∑ n∈S 1/n = 1.

Using the Weyl transfomations induced by diffeomorphisms we set up a cohomological problem for the Fefferman-Graham coefficients. The cohomologically nontrivial solutions remove the ambiguity and give the nonlocal terms in the effective action responsible for the trace anomalies.

In 1964, Ronald Graham proved that there exist relatively prime natural numbers a and b such that the sequence {An} defined by An = An−1 +An−2 (n ≥ 2;A0 = a,A1 = b) contains no prime numbers, and constructed a 34-digit pair satisfying this condition. In 1990, Donald Knuth found a 17-digit pair satisfying the same conditions. That same year, noting an improvement to Knuth’s computation, Herbert ...

We study the linearization of the Dirichlet-to-Neumann map for Poincaré-Einstein metrics in even dimensions on an arbitrary compact manifold with boundary. By fixing a suitable gauge, we make the linearized Einstein equation elliptic. In this gauge the linearization of the Dirichlet-to-Neumann map appears as the scattering matrix for an elliptic operator of 0-type, modified by some differential...

We answer the following question of R. L. Graham: What is the discrepancy of the lexicographically-least binary de Bruijn sequence? Here, “discrepancy” refers to the maximum (absolute) difference between the number of ones and the number of zeros in any initial segment of the sequence. We show that the answer is Θ(2 log n/n).

A finite set X in some Euclidean space Rn is called Ramsey if for any k there is a d such that whenever Rd is k-coloured it contains a monochromatic set congruent to X. This notion was introduced by Erdős, Graham, Montgomery, Rothschild, Spencer and Straus, who asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. This question (made into a co...

For any variety X with an action of a split torus over a field, we establish a spectral sequence connecting the equivariant and the non-equivariant higher Chow groups of X . For X smooth and projective, we give an explicit formula relating the equivariant and the non-equivariant higher Chow groups. This in particular implies that the spectral sequence degenerates. These results also hold with r...