Constructive Lower Bounds on Classical Multicolor Ramsey Numbers

This paper studies lower bounds for classical multicolor Ramsey numbers, first by giving a short overview of past results, and then by presenting several general constructions establishing new lower bounds for many diagonal and off-diagonal multicolor Ramsey numbers. In particular, we improve several lower bounds for R_k(4) and R_k(5) for some small k, including 415 \u3c = R_3(5), 634 \u3c = R_4(4), 2721 \u3c = R_4(5), 3416 \u3c = R_5(4) and 26082 \u3c = R_5(5). Most of the new lower bounds are consequences of general constructions

Small Ramsey Numbers

We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values

Enumeration of three term arithmetic progressions in fixed density sets

Additive combinatorics is built around the famous theorem by Szemer\'edi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemer\'edi's theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a generalization of Szemer\'edi's theorem using methods from real algebraic geometry.Comment: 62 pages. Update v2: Corrected some references. Update v3: Incorporated feedbac

A sharp threshold for random graphs with a monochromatic triangle in every edge coloring

Let $\R$ be the set of all finite graphs $G$ with the Ramsey property that every coloring of the edges of $G$ by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let $G(n,p)$ be the random graph on $n$ vertices with edge probability $p$. We prove that there exists a function $\hat c=\hat c(n)$ with $0 0$, as $n$ tends to infinity Pr[G(n,(1-\eps)\hat c/\sqrt{n}) \in \R ] \to 0 and Pr [ G(n,(1+\eps)\hat c/\sqrt{n}) \in \R ] \to 1. A crucial tool that is used in the proof and is of independent interest is a generalization of Szemer\'edi's Regularity Lemma to a certain hypergraph setting.Comment: 101 pages, Final version - to appear in Memoirs of the A.M.

Some arithmetic Ramsey problems and inverse theorems

In this dissertation we study arithmetic Ramsey type problems and inverse problems, in various settings. This work consists of two parts. In Part I, we study arithmetic Ramsey type problems over abelian groups. This part consists of three chapters. In Chapter 2, using hypergraph containers, we study the rainbow Erdos-Rothschild problem for sum-free sets. In Chapters 3 and 4, we study the avoidance density for (k,l)-sum-free sets. The upper bound constructions are given in Chapter 3, answering a question asked by Bajnok. We also improved the lower bound for infinitely many (k,l) in both chapters, and a special case of the sum-free conjecture is verified in Chapter 4. In Part II, we study inverse problems over nonabelian topological groups. Preliminaries to topological groups are given in Chapter 5. In Chapter 6, we first obtain classifications of connected groups and sets which satisfy the equality in Kemperman's inequality, answering a question asked by Kemperman in 1964. When the ambient group is compact, we also get a near equality version of the above result with a sharp exponent bound, which confirms conjectures by Griesmer and by Tao. A measure expansion gap result for simple Lie groups is also presented. In Chapter 7, we study the small measure expansion problem in noncompact locally compact groups. The question that whether there is a Brunn-Minkowski inequality was asked by Henstock and Macbeath in 1953. We obtain such an inequality and prove it is sharp for a large class of groups (including real linear algebraic groups, Nash groups, semisimple Lie groups with finite center, solvable Lie groups, etc), answering questions by Hrushovski and by Tao

Partition regularity of Pythagorean pairs

We address a core partition regularity problem in Ramsey theory by proving that every finite coloring of the positive integers contains monochromatic Pythagorean pairs, i.e., $x,y\in \mathbb{N}$ such that $x^2\pm y^2=z^2$ for some $z\in \mathbb{N}$. We also show that partitions generated by level sets of multiplicative functions taking finitely many values always contain Pythagorean triples. Our proofs combine known Gowers uniformity properties of aperiodic multiplicative functions with a novel and rather flexible approach based on concentration estimates of multiplicative functions.Comment: 45 page

Unraveling incomplete lupus:search for predictive markers for progression to systemic lupus erythematosus

Systemic lupus erythematosus (SLE) is a rare autoimmune disease that can be serious. There is no cure, but the disease can be slowed down with medication. The aim of this thesis is to gain more insight into the early phase of SLE. If recognized early, targeted treatment can be started in time to prevent organ damage. The researchers looked at the occurrence of early features of autoimmune diseases in the population, both for symptoms and antibodies. It shows that screening for specific autoantibodies in the population is only useful in a selected risk group, possibly on the basis of a questionnaire aimed at systemic diseases. Furthermore, research has been done on patients with features of SLE, without meeting the classification criteria, also known as incomplete SLE (iSLE). The patients were followed for a longer period of time, with the aim of distinguishing which patients have a high risk of progression to SLE. Based on this study and a literature analysis, there are several factors, namely female gender, younger age, increased IFN-type I activity, decreased complement and diversity of autoantibodies, which together suggest a higher risk of SLE. Therefore, we recommend repeat testing for antinuclear antibodies and serum complement concentrations in patients with iSLE. In the future, increased IFN-type I activity could possibly be investigated, using myxovirus resistance protein A as a surrogate marker, but these tests are not yet generally available. When changes in one or more of these factors occur, closer monitoring and considering early treatment is advised