In 1926 van der Waerden [13] proved the following startling theorem : If the set of integers is arbitrarily partitioned into two classes then at least one class contains arbitrarily long arithmetic progressions. It is well known and obvious that neither class must contain an infinite arithmetic progression. In fact, it is easy to see that for any sequence an there is another sequence bn9 with b...

We study quasi-random properties of k-uniform hypergraphs. Our central notion is uniform edge distribution with respect to large vertex sets. We will find several equivalent characterisations of this property and our work can be viewed as an extension of the well known Chung-Graham-Wilson theorem for quasi-random graphs. Moreover, let Kk be the complete graph on k vertices and M(k) the line gra...

which implies that the topology of the vorticity vector field must be very simple. Even local existence of the Clebsch representation is only guaranteed at points where the vorticity does not vanish. It has been shown (Graham-Henyey [17]) that in general (1.3) can not hold around a point with zero vorticity. For more discussion on the properties of classical Clebsch variables and its various ge...

We examine the problem of exactly or approximately counting all perfect matchings in hereditary classes of nonbipartite graphs. In particular, we consider the switch Markov chain of Diaconis, Graham and Holmes. We determine the largest hereditary class for which the chain is ergodic, and define a large new hereditary class of graphs for which it is rapidly mixing. We go on to show that the chai...

We construct a simple fully homomorphic encryption scheme, using only elementary modular arithmetic. We use Gentry’s technique to construct fully homomorphic scheme from a “bootstrappable” somewhat homomorphic scheme. However, instead of using ideal lattices over a polynomial ring, our bootstrappable encryption scheme merely uses addition and multiplication over the integers. The main appeal of...

For every n, we describe an explicit construction of a graph on n vertices with at most O(n2− ) edges, for = 0.133 . . ., that contains every graph on n vertices with maximum degree 3 as a subgraph. It is easy to see that each such graph must have at least Ω(n 4 3 ) edges. We also show that the minimum number of edges of a graph that contains every graph with n edges as a subgraph is Θ( n 2 log...

We construct error-correcting (nonlinear) binary codes using a construction of Bose and Chowla in additive number theory. Our method extends a construction of Graham and Sloane for constant weight codes. The new codes improve 1028 of the 7168 best known h-error correcting codes of wordlength ≤ 512 with 1 ≤ h ≤ 14. We give asymptotical comparisons to shortened BCH codes.

Chung and Graham conjectured (in 1981) that n points in the unit square [0, 1] can be connected by a rectilinear Steiner tree of length at most √ n+1. Here we confirm this conjecture for small values of n, and for some new infinite sequences of values of n (but not for all n). As an interesting byproduct we obtain close rational approximations of √ n from below, for those n.

Graham, Hoffman and Hosoya gave a very nice formula about the determinant of the distance matrix DG of a graph G in terms of the distance matrix of its blocks. We generalize this result to a q-analogue of DG. Our generalization yields results about the equality of the determinant of the mod-2 (and in general mod-k) distance matrix (i.e. each entry of the distance matrix is taken modulo 2 or k) ...

Scheduling algorithms for composite multimedia presentations need to ensure that the user-defined synchronization constraints for the various presentation components are met. This requirement gives rise to task models that are significantly more complex than the models employed in scheduling theory and practice. In this paper, we formulate the resource scheduling problems for composite multimed...