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

On certain other sets of integers

We show that if A is a subset of {1,...,N} containing no non-trivial three-term arithmetic progressions then |A|=O(N/ log^{3/4-o(1)} N).Comment: 29 pp. Corrected typos. Added definitions for some non-standard notation and remarks on lower bound

Label-free electrochemical monitoring of DNA ligase activity

This study presents a simple, label-free electrochemical technique for the monitoring of DNA ligase activity. DNA ligases are enzymes that catalyze joining of breaks in the backbone of DNA and are of significant scientific interest due to their essential nature in DNA metabolism and their importance to a range of molecular biological methodologies. The electrochemical behavior of DNA at mercury and some amalgam electrodes is strongly influenced by its backbone structure, allowing a perfect discrimination between DNA molecules containing or lacking free ends. This variation in electrochemical behavior has been utilized previously for a sensitive detection of DNA damage involving the sugar-phosphate backbone breakage. Here we show that the same principle can be utilized for monitoring of a reverse process, i.e., the repair of strand breaks by action of the DNA ligases. We demonstrate applications of the electrochemical technique for a distinction between ligatable and unligatable breaks in plasmid DNA using T4 DNA ligase, as well as for studies of the DNA backbone-joining activity in recombinant fragments of E. coli DNA ligase

Bounds for graph regularity and removal lemmas

We show, for any positive integer k, that there exists a graph in which any equitable partition of its vertices into k parts has at least ck^2/\log^* k pairs of parts which are not \epsilon-regular, where c,\epsilon>0 are absolute constants. This bound is tight up to the constant c and addresses a question of Gowers on the number of irregular pairs in Szemer\'edi's regularity lemma. In order to gain some control over irregular pairs, another regularity lemma, known as the strong regularity lemma, was developed by Alon, Fischer, Krivelevich, and Szegedy. For this lemma, we prove a lower bound of wowzer-type, which is one level higher in the Ackermann hierarchy than the tower function, on the number of parts in the strong regularity lemma, essentially matching the upper bound. On the other hand, for the induced graph removal lemma, the standard application of the strong regularity lemma, we find a different proof which yields a tower-type bound. We also discuss bounds on several related regularity lemmas, including the weak regularity lemma of Frieze and Kannan and the recently established regular approximation theorem. In particular, we show that a weak partition with approximation parameter \epsilon may require as many as 2^{\Omega(\epsilon^{-2})} parts. This is tight up to the implied constant and solves a problem studied by Lov\'asz and Szegedy.Comment: 62 page

Small doubling in groups

Let A be a subset of a group G = (G,.). We will survey the theory of sets A with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}. The case G = (Z,+) is the famous Freiman--Ruzsa theorem.Comment: 23 pages, survey article submitted to Proceedings of the Erdos Centenary conferenc

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

Towards model-based control of Parkinson's disease

Modern model-based control theory has led to transformative improvements in our ability to track the nonlinear dynamics of systems that we observe, and to engineer control systems of unprecedented efficacy. In parallel with these developments, our ability to build computational models to embody our expanding knowledge of the biophysics of neurons and their networks is maturing at a rapid rate. In the treatment of human dynamical disease, our employment of deep brain stimulators for the treatment of Parkinson’s disease is gaining increasing acceptance. Thus, the confluence of these three developments—control theory, computational neuroscience and deep brain stimulation—offers a unique opportunity to create novel approaches to the treatment of this disease. This paper explores the relevant state of the art of science, medicine and engineering, and proposes a strategy for model-based control of Parkinson’s disease. We present a set of preliminary calculations employing basal ganglia computational models, structured within an unscented Kalman filter for tracking observations and prescribing control. Based upon these findings, we will offer suggestions for future research and development

Exploring Pompeii: discovering hospitality through research synergy

Hospitality research continues to broaden through an ever-increasing dialogue and alignment with a greater number of academic disciplines. This paper demonstrates how an enhanced understanding of hospitality can be achieved through synergy between archaeology, the classics and sociology. It focuses on classical Roman life, in particular Pompeii, to illustrate the potential for research synergy and collaboration, to advance the debate on hospitality research and to encourage divergence in research approaches. It demonstrates evidence of commercial hospitality activities through the excavation hotels, bars and taverns, restaurants and fast food sites. The paper also provides an example of the benefits to be gained from multidisciplinary analysis of hospitality and tourism

Est locus uni cuique suus: City and Status in Horace’s Satires 1.8 and 1.9

This is the published version