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

Bipartite, Size, and Online Ramsey Numbers of Some Cycles and Paths

The basic premise of Ramsey Theory states that in a sufficiently large system, complete disorder is impossible. One instance from the world of graph theory says that given two fixed graphs F and H, there exists a finitely large graph G such that any red/blue edge coloring of the edges of G will produce a red copy of F or a blue copy of H. Much research has been conducted in recent decades on quantifying exactly how large G must be if we consider different classes of graphs for F and H. In this thesis, we explore several Ramsey- type problems with a particular focus on paths and cycles. We first examine the bipartite size Ramsey number of a path on n vertices, bˆr(Pn), and give an upper bound using a random graph construction motivated by prior upper bound improvements in similar problems. Next, we consider the size Ramsey number Rˆ (C, Pn) and provide a significant improvement to the upper bound using a very structured graph, the cube of a path, as opposed to a random construction. We also prove a small improvement to the lower bound and show that the r-colored version of this problem is asymptotically linear in rn. Lastly, we give an upper bound for the online Ramsey number R˜ (C, Pn)

Ramsey numbers involving a triangle: theory and algorithms

Ramsey theory studies the existence of highly regular patterns in large sets of objects. Given two graphs G and H, the Ramsey number R(G, H) is defined to be the smallest integer n such that any graph F with n or more vertices must contain G, or F must contain H. Albeit beautiful, the problem of determining Ramsey numbers is considered to be very difficult. We focus our attention on efficient algorithms for determining Ram sey numbers involving a triangle: R(K3 , G). With the help of theoretical tools, the search space is reduced by using different pruning techniques and linear programming. Efficient operations are also carried out to mathematically glue together small graphs to construct larger critical graphs. Using the algorithms developed in this thesis, we compute all the Ramsey numbers R(Kz,G), where G is any connected graph of order seven. Most of the corresponding critical graphs are also constructed. We believe that the algorithms developed here will have wider applications to other Ramsey-type problems

Tur\'an and Ramsey Properties of Subcube Intersection Graphs

The discrete cube $\{0,1\}^d$ is a fundamental combinatorial structure. A subcube of $\{0,1\}^d$ is a subset of $2^k$ of its points formed by fixing $k$ coordinates and allowing the remaining $d-k$ to vary freely. The subcube structure of the discrete cube is surprisingly complicated and there are many open questions relating to it. This paper is concerned with patterns of intersections among subcubes of the discrete cube. Two sample questions along these lines are as follows: given a family of subcubes in which no $r+1$ of them have non-empty intersection, how many pairwise intersections can we have? How many subcubes can we have if among them there are no $k$ which have non-empty intersection and no $l$ which are pairwise disjoint? These questions are naturally expressed as Tur\'an and Ramsey type questions in intersection graphs of subcubes where the intersection graph of a family of sets has one vertex for each set in the family with two vertices being adjacent if the corresponding subsets intersect. Tur\'an and Ramsey type problems are at the heart of extremal combinatorics and so these problems are mathematically natural. However, a second motivation is a connection with some questions in social choice theory arising from a simple model of agreement in a society. Specifically, if we have to make a binary choice on each of $n$ separate issues then it is reasonable to assume that the set of choices which are acceptable to an individual will be represented by a subcube. Consequently, the pattern of intersections within a family of subcubes will have implications for the level of agreement within a society. We pose a number of questions and conjectures relating directly to the Tur\'an and Ramsey problems as well as raising some further directions for study of subcube intersection graphs.Comment: 18 page

Upper density problems in infinite Ramsey theory

We consider the following question in infinite Ramsey theory, introduced by Erdős and Galvin [EG93] in a particular case and by DeBiasio and McKenney [DM19] in a more general setting. Let H be a countably infinite graph. If the edges of the complete graph on the natural numbers are colored red or blue, what is the maximum value of λ such that we are guaranteed to find a monochromatic copy of H whose vertex set has upper density at least λ? We call this value the Ramsey density of H. The problem of determining the Ramsey density of the infinite path was first studied by Erdős and Galvin, and was recently solved by Corsten, DeBiasio, Lang and the author [CDLL19]. In this thesis we study the problem of determining the Ramsey density of arbitrary graphs H. On an intuitive level, we show that three properties of a graph H have an effect on the Ramsey density: the chromatic number, the number of components, and the expansion of its independent sets. We deduce the exact value of the Ramsey density for a wide variety of graphs, including all locally finite forests, bipartite factors, clique factors and odd cycle factors. We also determine the value of the Ramsey density of all locally finite graphs, up to a factor of 2. We also study a list coloring variant of the same problem. We show that there exists a way of assigning a list of size two to every edge in the complete graph on N such that, in every list coloring, there are monochromatic paths with density arbitrarily close to 1.Wir betrachten die folgende Fragestellung aus der Ramsey-Theorie, welche von Erdős und Galvin [EG93] in einem Spezialfall sowie von DeBiasio und McKenney [DM19] in einem allgemeineren Kontext formuliert wurde: Es sei H ein abzählbar unendlicher Graph. Welches ist der größtmögliche Wert λ, sodass wir, wenn die Kanten des vollständigen Graphen mit Knotenmenge N jeweils entweder rot oder blau gefärbt sind, stets eine einfarbige Kopie von H, dessen Knotenmenge eine obere asymptotische Dichte von mindestens λ besitzt, finden können? Wir nennen diesen Wert die Ramsey-Dichte von H. Das Problem, die Ramsey-Dichte des unendlichen Pfades zu bestimmen wurde erstmals von Erdős und Galvin untersucht und wurde vor kurzem von Corsten, DeBiasio, Lang und dem Autor [CDLL19] gelöst. Gegenstand der vorliegenden Dissertation ist die Bestimmung der Ramsey-Dichten von Graphen. Auf einer intuitiven Ebene zeigen wir, dass drei Parameter eines Graphen die Ramsey-Dichte beeinflussen: die chromatische Zahl, die Anzahl der Zusammenhangskomponenten sowie die Expansion seiner unabhängigen Mengen. Wir ermitteln die exakten Werte der Ramsey-Dichte für eine Vielzahl von Graphen, darunter alle lokal endlichen Wälder, bipartite Faktoren, Kr-Faktoren sowie Ck-Faktoren für ungerade k. Ferner bestimmen wir den Wert der Ramsey-Dichte aller lokal endlichen Graphen bis auf einen Faktor 2. Darüber hinaus untersuchen wir eine Variante des oben beschriebenen Problems für Listenfärbungen. Wir zeigen, dass es möglich ist, jeder Kante des vollständigen Graphen mit Knotenmenge N eine Liste der Größe Zwei zuzuweisen, sodass in jeder zugehörigen Listenfärbung monochromatische Pfade mit beliebig nah an 1 liegender Dichte existieren

On Ramsey Theory and Slow Bootstrap Percolation

This dissertation concerns two sets of problems in extremal combinatorics. The major part, Chapters 1 to 4, is about Ramsey-type problems for cycles. The shorter second part, Chapter 5, is about a problem in bootstrap percolation. Next, we describe each topic more precisely. Given three graphs G, L1 and L2, we say that G arrows (L1, L2) and write G → (L1, L2), if for every edge-coloring of G by two colors, say 1 and 2, there exists a color i whose color class contains Li as a subgraph. The classical problem in Ramsey theory is the case where G, L1 and L2 are complete graphs; in this case the question is how large the order of G must be (in terms of the orders of L1 andL2) to guarantee that G → (L1, L2). Recently there has been much interest in the case where L1 and L2 are cycles and G is a graph whose minimum degree is large. In the past decade, numerous results have been proved about those problems. We will continue this work and prove two conjectures that have been left open. Our main weapon is Szemeredi\u27s Regularity Lemma.Our second topic is about a rather unusual aspect of the fast expanding theory of bootstrap percolation. Bootstrap percolation on a graph G with parameter r is a cellular automaton modeling the spread of an infection: starting with a set A0, cointained in V(G), of initially infected vertices, define a nested sequence of sets, A0 ⊆ A1 ⊆. . . ⊆ V(G), by the update rule that At+1, the set of vertices infected at time t + 1, is obtained from At by adding to it all vertices with at least r neighbors in At. The initial set A0 percolates if At = V(G) for some t. The minimal such t is the time it takes for A0 to percolate. We prove results about the maximum percolation time on the two-dimensional grid with parameter r = 2

Solving Hard Graph Problems with Combinatorial Computing and Optimization

Many problems arising in graph theory are difficult by nature, and finding solutions to large or complex instances of them often require the use of computers. As some such problems are $NP$-hard or lie even higher in the polynomial hierarchy, it is unlikely that efficient, exact algorithms will solve them. Therefore, alternative computational methods are used. Combinatorial computing is a branch of mathematics and computer science concerned with these methods, where algorithms are developed to generate and search through combinatorial structures in order to determine certain properties of them. In this thesis, we explore a number of such techniques, in the hopes of solving specific problem instances of interest. Three separate problems are considered, each of which is attacked with different methods of combinatorial computing and optimization. The first, originally proposed by ErdH{o}s and Hajnal in 1967, asks to find the Folkman number $F_e(3,3;4)$, defined as the smallest order of a $K_4$-free graph that is not the union of two triangle-free graphs. A notoriously difficult problem associated with Ramsey theory, the best known bounds on it prior to this work were $19 leq F_e(3,3;4) leq 941$. We improve the upper bound to $F_e(3,3;4) leq 786$ using a combination of known methods and the Goemans-Williamson semi-definite programming relaxation of MAX-CUT. The second problem of interest is the Ramsey number $R(C_4,K_m)$, which is the smallest $n$ such that any $n$-vertex graph contains a cycle of length four or an independent set of order $m$. With the help of combinatorial algorithms, we determine $R(C_4,K_9)=30$ and $R(C_4,K_{10})=36$ using large-scale computations on the Open Science Grid. Finally, we explore applications of the well-known Lenstra-Lenstra-Lov\u27{a}sz (LLL) algorithm, a polynomial-time algorithm that, when given a basis of a lattice, returns a basis for the same lattice with relatively short vectors. The main result of this work is an application to graph domination, where certain hard instances are solved using this algorithm as a heuristic

Topics in graph colouring and graph structures

This thesis investigates problems in a number of different areas of graph theory. These problems are related in the sense that they mostly concern the colouring or structure of the underlying graph. The first problem we consider is in Ramsey Theory, a branch of graph theory stemming from the eponymous theorem which, in its simplest form, states that any sufficiently large graph will contain a clique or anti-clique of a specified size. The problem of finding the minimum size of underlying graph which will guarantee such a clique or anti-clique is an interesting problem in its own right, which has received much interest over the last eighty years but which is notoriously intractable. We consider a generalisation of this problem. Rather than edges being present or not present in the underlying graph, each is assigned one of three possible colours and, rather than considering cliques, we consider cycles. Combining regularity and stability methods, we prove an exact result for a triple of long cycles. We then move on to consider removal lemmas. The classic Removal Lemma states that, for n sufficiently large, any graph on n vertices containing o(n^3) triangles can be made triangle-free by the removal of o(n^2) edges. Utilising a coloured hypergraph generalisation of this result, we prove removal lemmas for two classes of multinomials. Next, we consider a problem in fractional colouring. Since finding the chromatic number of a given graph can be viewed as an integer programming problem, it is natural to consider the solution to the corresponding linear programming problem. The solution to this LP-relaxation is called the fractional chromatic number. By a probabilistic method, we improve on the best previously known bound for the fractional chromatic number of a triangle-free graph with maximum degree at most three. Finally, we prove a weak version of Vizing's Theorem for hypergraphs. We prove that, if H is an intersecting 3-uniform hypergraph with maximum degree D and maximum multiplicity m, then H has at most 2D+m edges. Furthermore, we prove that the unique structure achieving this maximum is m copies of the Fano Plane

Continuous optimisation in extremal combinatorics

In this thesis we explore instances in which tools from continuous optimisation can be used to solve problems in extremal graph and hypergraph theory. We begin by introducing a generalised notion of hypergraph Lagrangian and use tools from the theory of nonlinear optimisation to explore some of its properties. As an application we find the Tur´an density of a small family of hypergraphs. We determine the exact k-colour Ramsey number of an odd cycle on n vertices when n is large. This resolves a conjecture of Bondy and Erd˝os for large n. The first step of our proof is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimisation. We establish a correspondence between extremal constructions in the Ramsey setting and optimal points in the continuous setting. We thereby uncover a correspondence between extremal constructions and perfect matchings in the k-dimensional hypercube. This allows us to prove a stability type result around these extremal constructions. We consider two models from statistical physics, the hard-core model and the monomer-dimer model. Using tools from linear programming we give tight upper bounds on the logarithmic derivative of the independence and matching polynomials of a d-regular graph. For independent sets, this is a strengthening of a sequence of results of Kahn, Galvin and Tetali, and Zhao that a disjoint union of Kd,d’s maximises the independence polynomial and total number of independent sets among all d-regular graphs on the same number of vertices. For matchings, the result implies that disjoint unions of Kd,d’s also maximise the matching polynomial and total number of matchings. Moreover we prove the Asymptotic Upper Matching Conjecture of Friedland, Krop, Lundow, and Markstr¨om. Through our study of the hard-core model, we also prove lower bounds on the average size and the number of independent sets in a triangle-free graph of maximum degree d. As a consequence we obtain a new proof of Shearer’s celebrated upper bound on the Ramsey number R(3, k)