For an undirected graph G = (V; E), let G n denote the graph whose vertex set is V n in which two distinct vertices (u 1 ; u 2 ; : : : ; un ) and (v 1 ; v 2 ; : : : ; v n ) are adjacent iff for all i between 1 and n either u i = v i or u i v i 2 E. The Shannon capacity c(G) of G is the limit limn7!1 (ff(G n )) 1=n , where ff(G n ) is the maximum size of an independent set of vertices in G n . We show that there are graphs G and H such that the Shannon capacity of their disjoint union is (much) bigger than the sum of their capacities. This disproves a conjecture of Shannon raised in 1956. 1 Introduction For an undirected graph G = (V; E), let G n denote the graph whose vertex set is V n in which two distinct vertices (u 1 ; u 2 ; : : : ; u n ) and (v 1 ; v 2 ; : : : ; v n ) are adjacent iff for all i between 1 and n either u i = v i or u i v i 2 E. The Shannon capacity c(G) of G is the limit lim n7!1 (ff(G n )) 1=n , where ff(G n ) is the maximum size of an inde...

We calculate the triple correlations for the truncated divisor sum λR(n). The λR(n) behave over certain averages just as the prime counting von Mangoldt function Λ(n) does or is conjectured to do. We also calculate the mixed (with a factor of Λ(n)) correlations. The results for the moments up to the third degree, and therefore the implications for the distribution of primes in short intervals, are the same as those we obtained (in the first paper with this title) by using the simpler approximation ΛR(n). However, when λR(n) is used, the error in the singular series approximation is often much smaller than what ΛR(n) allows. Assuming the Generalized Riemann Hypothesis (GRH) for Dirichlet L-functions, we obtain an Ω±-result for the variation of the error term in the prime number theorem. Formerly, our knowledge under GRH was restricted to Ω-results for the absolute value of this variation. An important ingredient in the last part of this work is a recent result due to Montgomery and Soundararajan which makes it possible for us to dispense with a large error term in the evaluation of a certain singular series average. We believe that our results on the sums λR(n) and ΛR(n) can be employed in diverse problems concerning primes. 1.

The degree/diameter problem is to determine the largest graphs or digraphs of given maximum degree and given diameter. General upper bounds – called Moore bounds – for the order of such graphs and digraphs are attainable only for certain special graphs and digraphs. Finding better (tighter) upper bounds for the maximum possible number of vertices, given the other two parameters, and thus attacking the degree/diameter problem ‘from above’, remains a largely unexplored area. Constructions producing large graphs and digraphs of given degree and diameter represent a way of attacking the degree/diameter problem ‘from below’. This sur-vey aims to give an overview of the current state-of-the-art of the degree/diameter problem. We focus mainly on the above two streams of research. However, we could not resist mentioning also results on various related problems. These include considering Moore-like bounds for special types of graphs and digraphs, such as vertex-transitive, Cayley, planar, bipartite, and many others, on the one hand, and related properties such as connectivity, regularity, and surface embeddability, on

Let vt(d; 2) be the largest order of a vertex-transitive graph of degree d and diameter two. It is known that vt(d; 2) = d 2 + 1 for d = 1; 2; 3, and 7; for the remaining values of d we have vt(d; 2) d 2 \Gamma 1. The only known general lower bound on vt(d; 2), valid for all d, seems to be vt(d; 2) b d+2 2 cd d+2 2 e. Using voltage graphs, we construct a family of vertex-transitive non-Cayley graphs which shows that vt(d; 2) 8 9 (d + 1 2 ) 2 for all d of the form d = (3q \Gamma 1)=2 where q is a prime power congruent with 1 (mod 4). The construction generalizes to all prime powers and yields large highly symmetric graphs for other degrees as well. In particular, for d = 7 we obtain as a special case the Hoffman-Singleton graph, and for d = 11 and d = 13 we have new largest graphs of diameter two and degree d on 98 and 162 vertices, respectively. 1 Introduction The well-known degree/diameter problem asks for determining the largest possible number n(d; k) of vertic...

Introduction The square G 2 of a graph G = (V; E) is the graph whose vertex set is V in which two distinct vertices are adjacent if and only if their distance in G is at most 2. What is the maximum possible chromatic number of G 2 , as G ranges over all graphs with maximum degree d and girth g? Our (somewhat surprising) answer is that for g = 3; 4; 5 or 6 this maximum is (1 + o(1))d 2 (where the o(1) term tends to 0 as d tends to infinity), whereas for all g 7, this maximum is of order d 2 = log d. To state this result more precisely, define, for every two integers d 2 and g<F9.9

In early 2005, Dan Goldston, János Pintz, and Cem Yıldırım [12] made a spectacular breakthrough in the study of prime numbers. Resolving a long-standing open problem, they proved that there are infinitely many primes for which the gap to the next prime is as small as we want compared to the average gap between

ABSTRACT. We use short divisor sums to approximate prime tuples and moments for primes in short intervals. By connecting these results to classical moment problems we are able to prove that, for any η> 0, a positive proportion of consecutive primes are within 1 + η times the average spacing between primes. 4 1.

In this paper, we prove that there exists a function ρk = (4 + o(1))k such that G(n, ρ/n) contains a k-regular graph with high probability whenever ρ> ρk. In the case of k = 3, it is also shown that G(n, ρ/n) contains a 3-regular graph with high probability whenever ρ> λ ≈ 5.1494. These are the first constant bounds on the average degree in G(n, p) for the existence of a k-regular subgraph. We also discuss the appearance of 3-regular subgraphs in cores of random graphs. 1

In this paper, we study r-uniform hypergraphs without cycles of length less than five, employing the definition of a hypergraph cycle due to Berge. In particular, for r = 3, we show that if has n vertices and a maximum number of edges, then 1 ). This also asymptotically determines the generalized Turan number T 3 (n, 8, 4). Some results are based on our bounds for the maximum size of Sidon-type sets in n .

Abstract. We examine an idea for approximating prime tuples. 1. Statement of results (Preliminary) In the present work we will prove the following result. Let pn denote the nth prime. Then (pn+1 − pn) (1.1) lim inf n→ ∞ log pn(log log pn) −1 < ∞. log log log log pn Further we show that supposing the validity of the Bombieri–Vinogradov theorem up to Q ≤ Xϑ with any level ϑ>1/2 we have bounded differences between consecutive primes infinitely often: (1.2) lim inf n→ ∞ (pn+1 − pn) ≤ C(ϑ) with a constant C(ϑ) depending only on ϑ. If the Bombieri–Vinogradov theorem holds with a level ϑ>20/21, in particular if the Elliott–Halberstam conjecture holds, then we obtain (1.3) lim inf n→ ∞ (pn+1 − pn) ≤ 20, that is pn+1 − pn ≤ 20 for infinitely many n. Inequalities (1.2)–(1.3) will follow from the even stronger following result Theorem A. Suppose the Bombieri–Vinogradov theorem is true for Q ≤ Xϑ with some ϑ>1/2. Then there exists a constant C ′ (ϑ) such that any admissible k-tuple contains at least two primes for any (1.4) k ≥ C ′ (ϑ) if ϑ>1/2, where C ′ (ϑ) is an explicitly calculable constant depending only on ϑ. Further we have at least two primes for (1.5) k =7 if ϑ>20/21. Remark. For the definition of admissibility see (2.2) below. We will show some more general results for the quantity (ν is a given positive integer) (1.6) Eν = lim inf n→∞ pn+ν − pn log pn