We shall be interested in the following Erdős-Ko-Rado-type question. Fix some set B ⊂ [n]. How large a family A ⊂ P [n] can we find such that the intersection of any two sets in A contains a cyclic translate (modulo n) of B? Chung, Graham, Frankl and Shearer have proved that, in the case where B is a block of length t, we can do no better than to take A to consist of all supersets of B. We give...

The notion of forcing pairs is located in the study of quasi-random graphs. Roughly speaking, a pair of graphs (F, F ′) is called forcing if the following holds: Suppose for a sequence of graphs (Gn) there is a p > 0 such that the number of copies of F and the number of copies of F ′ in every graph Gn of the sequence (Gn) is approximately the same as the expected value in the random graph G(n, ...

Graham et al. [3] proved a very attractive theorem about the determinant of the distance matrix DG of a connected graph G as a function of the distance matrix of its 2-connected blocks. In a connected graph, the distance between two vertices d(u, v) is the length of the shortest path between them. Let A be an n n matrix. Recall that for 1 i, j n, the cofactor ci,j is defined as ( 1) iþj times t...

In this work we re-examine two common modulus attacks on RSA. First, we show that Guo’s continued fraction attack works much better in practice than previously expected. Given three instances of RSA with a common modulus N and private exponents each smaller than N the attack can factor the modulus about 93% of the time in practice. The success rate of the attack can be increased up to almost 10...

To say that R is a root set modulo n means that R is a subset of Z n , the ring of integers modulo n , and there is a polynomial the roots of which modulo n are exactly the elements of R . Note that [ and Z n are always root sets modulo n . It seems that only two papers have appeared which mention the nature of root sets modulo n , and then only at a very basic level : Sierpin ́ ski [3] and Choj...

Let s(x) denote the maximum number of non-overlapping unit squares which can be packed into a large square of side length x. Let W (x) = x − s(x) denote the “wasted” area, i.e., the area not covered by the unit squares. In this note we prove that W (x) = O ( x √ 2)/7 log x ) . This improves earlier results of Erdős-Graham and Montgomery in which the upper bounds of W (x) = O(x) and W (x) = O(x(...

An old conjecture of Graham stated that if [Formula: see text] is a prime and [Formula: see text] is a sequence of [Formula: see text] terms from the cyclic group [Formula: see text] such that all (nontrivial) zero-sum subsequences have the same length, then [Formula: see text] must contain at most two distinct terms. In 1976, Erdős and Szemerédi gave a proof of the conjecture for sufficiently ...

The famous Graham-Pollak Theorem states that one needs at least n− 1 complete bipartite subgraphs to partition the edge set of the complete graph on n vertices. Originally proved in conjunction with addressing for networking problems, this theorem is also related to perfect hashing and various questions about communication complexity. Since it’s original proof using Sylvester’s Law of Intertia,...

If F is a family of graphs then the Turán density of F is determined by the minimum chromatic number of the members of F . The situation for Turán densities of 3-graphs is far more complex and still very unclear. Our aim in this paper is to present new exact Turán densities for individual and finite families of 3-graphs, in many cases we are also able to give corresponding stability results. As...

Haviland and Thomason and Chung and Graham were the rst to investigate systematically some properties of quasi-random hy-pergraphs. In particular, in a series of articles, Chung and Graham considered several quite disparate properties of random-like hypergraphs of density 1=2 and proved that they are in fact equivalent. The central concept in their work turned out to be the so called deviation ...