For a fixed r, let fr(n) denote the minimum number of complete r-partite rgraphs needed to partition the complete r-graph on n vertices. The Graham-Pollak theorem asserts that f2(n) = n − 1. An easy construction shows that fr(n) 6 (1 + o(1)) ( n br/2c ) , and we write cr for the least number such that fr(n) 6 cr(1 + o(1)) ( n br/2c ) . It was known that cr < 1 for each even r > 4, but this was ...

Graham, Jan Karel Lenstra, and Robert E. Tarjan Discrete.This article is about the large number named after Ronald Graham. While Graham was trying to explain a result in Ramsey theory which he had derived with his collaborator Bruce Lee. Ramseys Theorem for n-Parameter Sets PDF.on Ramsey theory. Szemerédis most famous theorem is at the heart of Ramsey theory. Graham is the informal administrato...

We prove a generalization of Graham’s Conjecture for optimal pebbling with arbitrary sets of target distributions. We provide bounds on optimal pebbling numbers of products of complete graphs and explicitly find optimal t-pebbling numbers for specific such products. We obtain bounds on optimal pebbling numbers of powers of the cycle C5. Finally, we present explicit distributions which provide a...

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...

The aim of this paper is the determination of the largest n-dimensional polytope with n+3 vertices of unit diameter. This is a special case of a more general problem Graham proposes in [2].

Let g(n) denote the least integer such that among any g(n) points in general position in the plane there are always n in convex position. In 1935 P. Erd} os and G. Szekeres showed that g(n) exists and 2 n?2 + 1 g(n) 2n?4 n?2 + 1. Recently, the upper bound has been slightly improved by Chung and Graham and by Kleitman and Pachter. In this note we further improve the upper bound to

Graham and Pollak [Bell System Tech. J. 50 (1971) 2495–2519] obtained a beautiful formula on the determinant of distance matrices of trees, which is independent of the structure of the trees. In this paper we give a simple proof of Graham and Pollak’s result. © 2005 Elsevier Inc. All rights reserved.

Erdös and Turán once conjectured that any set A ⊂ N with

In fact, this statement is a corollary to a more general theorem on well-partiallyordered sets. Here a partially ordered set is called well-partially-ordered, if every non-empty subset has at least one, but no more than a finite number of minimal elements (finite basis property). For instance, the set A∗, where A is a finite alphabet, under the scattered subword relation ≤, i.e., v ≤ w if and o...