We investigate Ramsey numbers of bounded degree graphs and provide an interpolation between known results on the Ramsey numbers of general bounded degree graphs and bounded degree graphs of small bandwidth. Our main theorem implies that there exists a constant c such that for every ∆, there exists β such that if G is an n-vertex graph with maximum degree at most ∆ having a homomorphism f into a...

We prove that the product replacement graph on generating 3-tuples of An is connected for n ≤ 11. We employ an efficient heuristic based on [P1] which works significantly faster than brute force. The heuristic works for any group. Our tests were confined to An due to the interest in Wiegold’s Conjecture, usually stated in terms of T -systems (see [P2]). Our results confirm Wiegold’s Conjecture ...

This is a strange and ultimately unsatisfying book. It is an examination of the self-image created by the slogan “It’s all in the genes.” One reason that the analysis is ultimately unsatisfying is due to Graham’s decision to focus his philosophical analysis on popular expositions of genetics and biotechnology and the way in which these expositions shape a popular understanding of the implicatio...

In this expository note, we discuss the celebrated theorem known as “van der Waerden’s theorem on arithemetic progressions,” the history of work on upper and lower bounds for the function associated with this theorem, a number of generalizations, and some open problems. 1 van der Waerden’s theorem, and the function w(k) The famous theorem of van der Waerden on arithmetic progressions is usually...

Let f be an edge ordering of Kn: a bijection E(Kn) → {1, 2, . . . , ( n 2 ) }. For an edge e ∈ E(Kn), we call f(e) the label of e. An increasing path in Kn is a simple path (visiting each vertex at most once) such that the label on each edge is greater than the label on the previous edge. We let S(f) be the number of edges in the longest increasing path. Chvátal and Komlós raised the question o...

The wish to determine the complete set of rational integral solutions of (1) was expressed by Diaconis and Graham in [2, p. 328]. Apparently, the solutions to this diophantine problem correspond to values of keN for which the Radon transform based on the set of all xeZ\ with exactly four ones is not invertible. We thank Hendrik Lenstra who communicated the problem to Jaap Top, to whom we are eq...

We consider the problem of computing the second elementary symmetric polynomial S2 n(X) ∆ = ∑ 1≤i<j≤n XiXj using depth-three arithmetic circuits of the form ∑r i=1 ∏si j=1 Lij(X), where each Lij is a linear form in X1, . . . ,Xn. We consider this problem over several fields and determine exactly the number of multiplication gates required. The lower bounds are proved for inhomogeneous circuits ...

A partial cube is a graph that can be isometrically embedded into a hypercube. In other words, a partial cube is a subgraph of a hypercube that preserves distances the distance between any two vertices in the subgraph is the same as the distance between those vertices in the hypercube. Partial cubes were first introduced by Graham and Pollak (Graham & Pollak, 1971) as a model for communication ...

We present evidence for a strong correlation between the concentration of bulges and the mass of their central supermassive black hole (Mbh) — more concentrated bulges have more massive black holes. Using Cre(1/3) from Trujillo, Graham, & Caon (2001b) as a measure of bulge concentration, we find that log(Mbh/M⊙) = 6.81(±0.95)Cre(1/3)+ 5.03± 0.41. This correlation is shown to be marginally stron...

[Gr72] Graham R., " An efficient algorithm for determining the convex hull of a planar set, " In-A simple algorithm for determining the envelope of a set of lines, " Information [To91] Toussaint G.T., " Computing morphological properties of arrangements of lines, " Proc. [Ve87] Vegter G., " Computing the bounded region determined by finitely many lines in the plane, " Technical Report 87-03, [Z...