A review is given of some of the mathematical research of Bruce Rothschild, emphasizing his results in combinatorial theory, especially that part known as Ramsey Theory. Special emphasis is given to the Graham-Rothschild Parameter Sets Theorem, its consequences, and some extensions.

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 ...

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 this paper we provide bounds for the size of the solutions of the Diophantine equation x(x+ 1)(x+ 2)(x+ 3)(x+ k)(x+ k+ 1)(x+ k+ 2)(x+ k+ 3) = y, where 4 ≤ k ∈ N is a parameter. We also determine all integral solutions for 1 ≤ k ≤ 10.