Vacuum polarization radiative correction to the parity violating electron scattering on heavy nuclei

The effect of vacuum polarization on the parity violating asymmetry in the elastic electron-nucleus scattering is considered. Calculations are performed in the high-energy approximation with an exact account for the electric field of the nucleus. It is shown that the radiative correction to the parity violating asymmetry is logarithmically enhanced and the value of the correction is about -1%.Comment: 6 pages, 3 figures, REVTex

Counting Hamilton cycles in sparse random directed graphs

Let D(n,p) be the random directed graph on n vertices where each of the n(n-1) possible arcs is present independently with probability p. A celebrated result of Frieze shows that if $p\ge(\log n+\omega(1))/n$ then D(n,p) typically has a directed Hamilton cycle, and this is best possible. In this paper, we obtain a strengthening of this result, showing that under the same condition, the number of directed Hamilton cycles in D(n,p) is typically $n!(p(1+o(1)))^{n}$. We also prove a hitting-time version of this statement, showing that in the random directed graph process, as soon as every vertex has in-/out-degrees at least 1, there are typically $n!(\log n/n(1+o(1)))^{n}$ directed Hamilton cycles

On a problem of Erd\H{o}s and Rothschild on edges in triangles

Erd\H{o}s and Rothschild asked to estimate the maximum number, denoted by H(N,C), such that every N-vertex graph with at least CN^2 edges, each of which is contained in at least one triangle, must contain an edge that is in at least H(N,C) triangles. In particular, Erd\H{o}s asked in 1987 to determine whether for every C>0 there is \epsilon >0 such that H(N,C) > N^\epsilon, for all sufficiently large N. We prove that H(N,C) = N^{O(1/log log N)} for every fixed C < 1/4. This gives a negative answer to the question of Erd\H{o}s, and is best possible in terms of the range for C, as it is known that every N-vertex graph with more than (N^2)/4 edges contains an edge that is in at least N/6 triangles.Comment: 8 page

On two problems in graph Ramsey theory

We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices. The Ramsey number r(H) of a graph H is the least positive integer N such that every two-coloring of the edges of the complete graph $K_N$ contains a monochromatic copy of H. A famous result of Chv\'atal, R\"{o}dl, Szemer\'edi and Trotter states that there exists a constant c(\Delta) such that r(H) \leq c(\Delta) n for every graph H with n vertices and maximum degree \Delta. The important open question is to determine the constant c(\Delta). The best results, both due to Graham, R\"{o}dl and Ruci\'nski, state that there are constants c and c' such that 2^{c' \Delta} \leq c(\Delta) \leq 2^{c \Delta \log^2 \Delta}. We improve this upper bound, showing that there is a constant c for which c(\Delta) \leq 2^{c \Delta \log \Delta}. The induced Ramsey number r_{ind}(H) of a graph H is the least positive integer N for which there exists a graph G on N vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of H. Erd\H{o}s conjectured the existence of a constant c such that, for any graph H on n vertices, r_{ind}(H) \leq 2^{c n}. We move a step closer to proving this conjecture, showing that r_{ind} (H) \leq 2^{c n \log n}. This improves upon an earlier result of Kohayakawa, Pr\"{o}mel and R\"{o}dl by a factor of \log n in the exponent.Comment: 18 page

The early evolution of the H-free process

The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as $n \to \infty$, the minimum degree in G is at least $cn^{1-(v_H-2)/(e_H-1)}(\log n)^{1/(e_H-1)}$. This gives new lower bounds for the Tur\'an numbers of certain bipartite graphs, such as the complete bipartite graphs $K_{r,r}$ with $r \ge 5$. When H is a complete graph $K_s$ with $s \ge 5$ we show that for some C>0, with high probability the independence number of G is at most $Cn^{2/(s+1)}(\log n)^{1-1/(e_H-1)}$. This gives new lower bounds for Ramsey numbers R(s,t) for fixed $s \ge 5$ and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.Comment: 36 page

Maximum union-free subfamilies

An old problem of Moser asks: how large of a union-free subfamily does every family of m sets have? A family of sets is called union-free if there are no three distinct sets in the family such that the union of two of the sets is equal to the third set. We show that every family of m sets contains a union-free subfamily of size at least \lfloor \sqrt{4m+1}\rfloor - 1 and that this bound is tight. This solves Moser's problem and proves a conjecture of Erd\H{o}s and Shelah from 1972. More generally, a family of sets is a-union-free if there are no a+1 distinct sets in the family such that one of them is equal to the union of a others. We determine up to an absolute multiplicative constant factor the size of the largest guaranteed a-union-free subfamily of a family of m sets. Our result verifies in a strong form a conjecture of Barat, F\"{u}redi, Kantor, Kim and Patkos.Comment: 10 page

A model of a transition neutral pion formfactor measured in annihilation and scattering channels

We consider an alternative explanation of newly found growth of neutral pion transition form factor with virtuality of one of photon. It is based on Sudakov suppression of quark-photon vertex. Some applications to scattering and annihilation channels are considered including the relevant experiments with lepton-proton scattering.Comment: 5 pages, 4 figur

Worldline Approach to Forward and Fixed Angle fermion-fermion Scattering in Yang-Mills Theories at High Energies

Worldline techniques are employed to study the general behaviour of the fermion-fermion collision amplitude at very high energies in a non-abelian gauge field theory for the forward and fixed angle scattering cases. A central objective of this work is to demonstrate the simplicity by which the worldline methodology isolates that sector of the full theory which carries the soft physics, relevant to each process. Anomalous dimensions pertaining to a given soft sector are identified and subseuently used to facilitate the renormalization group running of the respective four point functions. Gluon reggeization is achieved for forward, while Sudakov suppression is established for fixed angle scattering.Comment: 28 pages, 10 figures in three file

Working directly with probabilities in quantum field theory

We present a novel approach to computing transition probabilities in quantum field theory, which allows them to be written directly in terms of expectation values of nested commutators and anti-commutators of field operators, rather than squared matrix elements. We show that this leads to a diagrammatic expansion in which the retarded propagator plays a dominant role. As a result, one is able to see clearly how faster-than-light signalling is prevented between sources and detectors. Finally, we comment on potential implications of this approach for dealing with infra-red divergences

The critical window for the classical Ramsey-Tur\'an problem

The first application of Szemer\'edi's powerful regularity method was the following celebrated Ramsey-Tur\'an result proved by Szemer\'edi in 1972: any K_4-free graph on N vertices with independence number o(N) has at most (1/8 + o(1)) N^2 edges. Four years later, Bollob\'as and Erd\H{o}s gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K_4-free graph on N vertices with independence number o(N) and (1/8 - o(1)) N^2 edges. Starting with Bollob\'as and Erd\H{o}s in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about N^2 / 8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.Comment: 34 page