Extremal results in sparse pseudorandom graphs

Szemer\'edi's regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and R\"odl proved an analogue of Szemer\'edi's regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemer\'edi's regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a well-known open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and R\"odl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several well-known combinatorial theorems, including the removal lemmas for graphs and groups, the Erd\H{o}s-Stone-Simonovits theorem and Ramsey's theorem. These results extend and improve upon a substantial body of previous work.Comment: 70 pages, accepted for publication in Adv. Mat

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

Induced Ramsey-type theorems

We present a unified approach to proving Ramsey-type theorems for graphs with a forbidden induced subgraph which can be used to extend and improve the earlier results of Rodl, Erdos-Hajnal, Promel-Rodl, Nikiforov, Chung-Graham, and Luczak-Rodl. The proofs are based on a simple lemma (generalizing one by Graham, Rodl, and Rucinski) that can be used as a replacement for Szemeredi's regularity lemma, thereby giving much better bounds. The same approach can be also used to show that pseudo-random graphs have strong induced Ramsey properties. This leads to explicit constructions for upper bounds on various induced Ramsey numbers.Comment: 30 page

Recent developments in graph Ramsey theory

Given a graph H, the Ramsey number r(H) is the smallest natural number N such that any two-colouring of the edges of K_N contains a monochromatic copy of H. The existence of these numbers has been known since 1930 but their quantitative behaviour is still not well understood. Even so, there has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics. In this survey, we will describe some of this progress

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

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

On \u3cem\u3eConvivencia\u3c/em\u3e, Bridges and Boundaries: Belonging and exclusion in the narratives of Spain’s Arab-Islamic past

References to the history of al-Andalus, the medieval Muslim territory of the Iberian Peninsula, in what is today the region of Andalusia (Spain) still have a palpable presence and relevance. This dissertation examines diverse accounts of the Arab-Islamic past, and the ways and contexts in which they are invoked. Based on a year and a half of fieldwork in Granada, Spain, I conducted interviews with ordinary Andalusians, academics and researchers (primarily historians), tour guides, historical novelists, high school history teachers, Spanish-born Muslim converts to Islam, Moroccans, and others involved in the contemporary production of this history. Moreover, I conducted participant observation at national and regional commemorations, celebrations and historical sites, areas where this ‘Moorish’ history, as it is commonly known, is a central feature. I argue that: (1) historical accounts of al-Andalus cannot be reduced to the two polarized versions (or “sides”) dominant in political discourse and in much academic debate – one that views the Reconquista as liberation and another that views it as a tragedy – rather, there is a broad and often neglected spectrum between these opposing versions; (2) Andalusia draws on the Arab-Islamic past to promote its tourist industry, and its economic, political and cultural relations with the Arab world. It is safe to suggest that Andalusia is pulled between a history that bridges Europe and the Arab world, and a contemporary European border that reminds us of contemporary geopolitical divisions and separations; (3) Andalusian history and historical sites are commodified to maintain revenue from the tourist industry. Yet, in the process, inhabitants of the Albayzin, the Moorish quarter, adopt similar tourist practices to learn about their own history and appropriate global heritage tourism discourse to contest governmental decisions that benefit tourists to the detriment of residents; (4) commemorations and celebrations in the city weave together a dominant narrative that reinforces the national narrative and its myth of origin; concurrently, these annual rituals provide spaces for alternative versions to circulate, including those that are opposed to the official versions. Importantly, the Día de la Toma (Day of the Capture) commemoration symbolizing national unity is the most publicly contested