Bounded VC-Dimension Implies the Schur-Erd?s Conjecture

In 1916, Schur introduced the Ramsey number r(3;m), which is the minimum integer n > 1 such that for any m-coloring of the edges of the complete graph K_n, there is a monochromatic copy of K?. He showed that r(3;m) ? O(m!), and a simple construction demonstrates that r(3;m) ? 2^?(m). An old conjecture of Erd?s states that r(3;m) = 2^?(m). In this note, we prove the conjecture for m-colorings with bounded VC-dimension, that is, for m-colorings with the property that the set system induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension

Turán problems in graphs and hypergraphs

Mantel's theorem says that among all triangle-free graphs of a given order the balanced complete bipartite graph is the unique graph of maximum size. In Chapter 2, we prove an analogue of this result for 3-graphs (3-uniform hy¬pergraphs) together with an associated stability result. Let K− 4 , F5 and F6 be 3-graphs with vertex sets {1, 2,3, 4}, {1, 2,3,4, 5} and {1, 2,3,4, 5, 6} re¬spectively and edge sets E(K−4 ) = {123, 124, 134}, E(F5) = {123, 124, 345}, E(F6) = {123, 124,345, 156} and F = {K4, F6}. For n =6 5 the unique F-free 3-graph of order n and maximum size is the balanced complete tri¬partite 3-graph S3(n). This extends an old result of Bollobas that S3(n) is the unique 3-graph of maximum size with no copy of K− 4 or F5. In 1941, Turán generalised Mantel's theorem to cliques of arbitrary size and then asked whether similar results could be obtained for cliques on hyper-graphs. This has become one of the central unsolved problems in the field of extremal combinatorics. In Chapter 3, we prove that the Turán density of K(3) 5 together with six other induced subgraphs is 3/4. This is analogous to a similar result obtained for K(3) 4 by Razborov. In Chapter 4, we consider various generalisations of the Turán density. For example, we prove that, if the density in C of ¯P3 is x and C is K3-free, then |E(C)| /(n ) ≤ 1/4+(1/4)J1 − (8/3)x. This is motivated by the observation 2 that the extremal graph for K3 is ¯P3-free, so that the upper bound is a natural extension of a stability result for K3. The question how many edges can be deleted from a blow-up of H before it is H-free subject to the constraint that the same proportion of edges are deleted from each connected pair of vertex sets has become known as the Turán density problem. In Chapter 5, using entropy compression supplemented with some analytic methods, we derive an upper bound of 1 − 1/('y(Δ(H) − /3)), where Δ(H) is the maximum degree of H, 3 ≤ 'y < 4 and /3 ≤ 1. The new bound asymptotically approaches the existing best upper bound despite being derived in a completely different way. The techniques used in these results, illustrating their breadth and connec¬tions between them, are set out in Chapter 1

Interactions between Ergodic Theory and Combinatorial Number Theory

The seminal work of Furstenberg on his ergodic proof of Szemerédi’s Theorem gave rise to a very rich connection between Ergodic Theory and Combinatorial Number Theory (Additive Combinatorics). The former is concerned with dynamics on probability spaces, while the latter is concerned with Ramsey theoretic questions about the integers, as well as other groups. This thesis further develops this symbiosis by establishing various combinatorial results via ergodic techniques, and vice versa. Let us now briefly list some examples of such. A shorter ergodic proof of the following theorem of Magyar is given: If B Zd, where d 5, has upper Banach density at least > 0, then the set of all squared distances in B, i.e., the set fkb1 b2k2 j b1; b2 2 Bg, contains qZ>R for some integer q = q( ) > 0 and R = R(B). Our technique also gives rise to results on the abundance of many other higher order Euclidean configurations in such sets. Next, we turn to establishing analogues of this result of Magyar, where k k2 is replaced with other quadratic forms and various other algebraic functions. Such results were initially obtained by Björklund and Fish, but their techniques involved some deep measure rigidity results of Benoist-Quint. We are able to recover many of their results and prove some completely new ones (not obtainable by their techniques) in a much more self-contained way by avoiding these deep results of Benoist-Quint and using only classical tools from Ergodic Theory. Finally, we extend some recent ergodic analogues of the classical Plünnecke inequalities for sumsets obtained by Björklund-Fish and establish some estimates of the Banach density of product sets in amenable non-abelain groups. We have aimed to make this thesis accesible to readers outside of Ergodic Theory who may be primarily interested in the arithmetic and combinatorial applications

Polynomial growth of concept lattices, canonical bases and generators:: extremal set theory in Formal Concept Analysis

We prove that there exist three distinct, comprehensive classes of (formal) contexts with polynomially many concepts. Namely: contexts which are nowhere dense, of bounded breadth or highly convex. Already present in G. Birkhoff's classic monograph is the notion of breadth of a lattice; it equals the number of atoms of a largest boolean suborder. Even though it is natural to define the breadth of a context as being that of its concept lattice, this idea had not been exploited before. We do this and establish many equivalences. Amongst them, it is shown that the breadth of a context equals the size of its largest minimal generator, its largest contranominal-scale subcontext, as well as the Vapnik-Chervonenkis dimension of both its system of extents and of intents. The polynomiality of the aforementioned classes is proven via upper bounds (also known as majorants) for the number of maximal bipartite cliques in bipartite graphs. These are results obtained by various authors in the last decades. The fact that they yield statements about formal contexts is a reward for investigating how two established fields interact, specifically Formal Concept Analysis (FCA) and graph theory. We improve considerably the breadth bound. Such improvement is twofold: besides giving a much tighter expression, we prove that it limits the number of minimal generators. This is strictly more general than upper bounding the quantity of concepts. Indeed, it automatically implies a bound on these, as well as on the number of proper premises. A corollary is that this improved result is a bound for the number of implications in the canonical basis too. With respect to the quantity of concepts, this sharper majorant is shown to be best possible. Such fact is established by constructing contexts whose concept lattices exhibit exactly that many elements. These structures are termed, respectively, extremal contexts and extremal lattices. The usual procedure of taking the standard context allows one to work interchangeably with either one of these two extremal structures. Extremal lattices are equivalently defined as finite lattices which have as many elements as possible, under the condition that they obey two upper limits: one for its number of join-irreducibles, other for its breadth. Subsequently, these structures are characterized in two ways. Our first characterization is done using the lattice perspective. Initially, we construct extremal lattices by the iterated operation of finding smaller, extremal subsemilattices and duplicating their elements. Then, it is shown that every extremal lattice must be obtained through a recursive application of this construction principle. A byproduct of this contribution is that extremal lattices are always meet-distributive. Despite the fact that this approach is revealing, the vicinity of its findings contains unanswered combinatorial questions which are relevant. Most notably, the number of meet-irreducibles of extremal lattices escapes from control when this construction is conducted. Aiming to get a grip on the number of meet-irreducibles, we succeed at proving an alternative characterization of these structures. This second approach is based on implication logic, and exposes an interesting link between number of proper premises, pseudo-extents and concepts. A guiding idea in this scenario is to use implications to construct lattices. It turns out that constructing extremal structures with this method is simpler, in the sense that a recursive application of the construction principle is not needed. Moreover, we obtain with ease a general, explicit formula for the Whitney numbers of extremal lattices. This reveals that they are unimodal, too. Like the first, this second construction method is shown to be characteristic. A particular case of the construction is able to force - with precision - a high number of (in the sense of "exponentially many'') meet-irreducibles. Such occasional explosion of meet-irreducibles motivates a generalization of the notion of extremal lattices. This is done by means of considering a more refined partition of the class of all finite lattices. In this finer-grained setting, each extremal class consists of lattices with bounded breadth, number of join irreducibles and meet-irreducibles as well. The generalized problem of finding the maximum number of concepts reveals itself to be challenging. Instead of attempting to classify these structures completely, we pose questions inspired by Turán's seminal result in extremal combinatorics. Most prominently: do extremal lattices (in this more general sense) have the maximum permitted breadth? We show a general statement in this setting: for every choice of limits (breadth, number of join-irreducibles and meet-irreducibles), we produce some extremal lattice with the maximum permitted breadth. The tools which underpin all the intuitions in this scenario are hypergraphs and exact set covers. In a rather unexpected, but interesting turn of events, we obtain for free a simple and interesting theorem about the general existence of "rich'' subcontexts. Precisely: every context contains an object/attribute pair which, after removed, results in a context with at least half the original number of concepts

Grafos com poucos cruzamentos e o número de cruzamentos do Kp,q em superfícies topológicas

Orientador: Orlando LeeTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O número de cruzamentos de um grafo G em uma superfície ? é o menor número de cruzamentos de arestas dentre todos os possíveis desenhos de G em ?. Esta tese aborda dois problemas distintos envolvendo número de cruzamentos de grafos: caracterização de grafos com número de cruzamentos igual a um e determinação do número de cruzamentos do Kp,q em superfícies topológicas. Para grafos com número de cruzamentos um, apresentamos uma completa caracterização estrutural. Também desenvolvemos um algoritmo "prático" para reconhecer estes grafos. Em relação ao número de cruzamentos do Kp,q em superfícies, mostramos que para um inteiro positivo p e uma superfície ? fixos, existe um conjunto finito D(p,?) de desenhos "bons" de grafos bipartidos completos Kp,r (possivelmente variando o r) tal que, para todo inteiro q e todo desenho D de Kp,q, existe um desenho bom D' de Kp,q obtido através de duplicação de vértices de um desenho D'' em D(p,?) tal que o número de cruzamentos de D' é menor ou igual ao número de cruzamentos de D. Em particular, para todo q suficientemente grande, existe algum desenho do Kp,q com o menor número de cruzamentos possível que é obtido a partir de algum desenho de D(p,?) através da duplicação de vértices do mesmo. Esse resultado é uma extensão de outro obtido por Cristian et. al. para esferaAbstract: The crossing number of a graph G in a surface ? is the least amount of edge crossings among all possible drawings of G in ?. This thesis deals with two problems on crossing number of graphs: characterization of graphs with crossing number one and determining the crossing number of Kp,q in topological surfaces. For graphs with crossing number one, we present a complete structural characterization. We also show a "practical" algorithm for recognition of such graphs. For the crossing number of Kp,q in surfaces, we show that for a fixed positive integer p and a fixed surface ?, there is a finite set D(p,?) of good drawings of complete bipartite graphs Kp,r (with distinct values of r) such that, for every positive integer q and every good drawing D of Kp,q, there is a good drawing D' of Kp,q obtained from a drawing D'' of D(p,?) by duplicating vertices of D'' and such that the crossing number of D' is at most the crossing number of D. In particular, for any large enough q, there exists some drawing of Kp,q with fewest crossings which can be obtained from a drawing of D(p,?) by duplicating vertices. This extends a result of Christian et. al. for the sphereDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação2014/14375-9FAPES