Colored complete hypergraphs containing no rainbow Berge triangles

The study of graph Ramsey numbers within restricted colorings, in particular forbidding a rainbow triangle, has recently been blossoming under the name Gallai-Ramsey numbers. In this work, we extend the main structural tool from rainbow triangle free colorings of complete graphs to rainbow Berge triangle free colorings of hypergraphs. In doing so, some other concepts and results are also translated from graphs to hypergraphs

Monochromatic kk-connected Subgraphs in 2-edge-colored Complete Graphs

Bollob\'{a}s and Gy\'{a}rf\'{a}s conjectured that for any $k, n \in \mathbb{Z}^+$ with $n > 4(k-1)$, every 2-edge-coloring of the complete graph on $n$ vertices leads to a $k$-connected monochromatic subgraph with at least $n-2k+2$ vertices. We find a counterexample with $n = 5k-2\lceil\sqrt{2k-1}\rceil-3$, thus disproving the conjecture, and we show the conjecture is true for $n \ge 5k-\min\{\sqrt{4k-2}+3, 0.5k+4\}$

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

Rainbow Generalizations of Ramsey Theory - A Dynamic Survey

In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs

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

Combinatorics

Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session

Extremal and Structural Problems of Graphs

In this dissertation, we are interested in studying several parameters of graphs and understanding their extreme values. We begin in Chapter~$2$ with a question on edge colouring. When can a partial proper edge colouring of a graph of maximum degree $\Delta$ be extended to a proper colouring of the entire graph using an `optimal' set of colours? Albertson and Moore conjectured this is always possible provided no two precoloured edges are within distance $2$. The main result of Chapter~$2$ comes close to proving this conjecture. Moreover, in Chapter~$3$, we completely answer the previous question for the class of planar graphs. Next, in Chapter~$4$, we investigate some Ramsey theoretical problems. We determine exactly what minimum degree a graph $G$ must have to guarantee that, for any two-colouring of $E(G)$, we can partition $V(G)$ into two parts where each part induces a connected monochromatic subgraph. This completely resolves a conjecture of Bal and Debiasio. We also prove a `covering' version of this result. Finally, we study another variant of these problems which deals with coverings of a graph by monochromatic components of distinct colours. The following saturation problem proposed by Barrus, Ferrara, Vandenbussche, and Wenger is considered in Chapter~$5$. Given a graph $H$ and a set of colours $\{1,2,\ldots,t\}$ (for some integer $t\geq |E(H)|$), we define $sat_{t}(n, R(H))$ to be the minimum number of $t$-coloured edges in a graph on $n$ vertices which does not contain a rainbow copy of $H$ but the addition of any non-edge in any colour from $\{1,2,\ldots,t\}$ creates such a copy. We prove several results concerning these extremal numbers. In particular, we determine the correct order of $sat_{t}(n, R(H))$, as a function of $n$, for every connected graph $H$ of minimum degree greater than $1$ and for every integer $t\geq e(H)$. In Chapter~$6$, we consider the following question: under what conditions does a Hamiltonian graph on $n$ vertices possess a second cycle of length at least $n-o(n)$? We prove that the `weak' assumption of a minimum degree greater or equal to $3$ guarantees the existence of such a long cycle. We solve two problems related to majority colouring in Chapter~$7$. This topic was recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised the problem of determining, for a natural number $k$, the smallest positive integer $m = m(k)$ such that every digraph can be coloured with $m$ colours, where each vertex has the same colour as at most a proportion of $\frac{1}{k}$ of its out-neighbours. Our main theorem states that $m(k) \in \{2k-1, 2k\}$. We study the following problem, raised by Caro and Yuster, in Chapter~$8$. Does every graph $G$ contain a `large' induced subgraph $H$ which has $k$ vertices of degree exactly $\Delta(H)$? We answer in the affirmative an approximate version of this question. Indeed, we prove that, for every $k$, there exists $g(k)$ such that any $n$ vertex graph $G$ with maximum degree $\Delta$ contains an induced subgraph $H$ with at least $n-g(k)\sqrt{\Delta}$ vertices such that $V(H)$ contains at least $k$ vertices of the same degree $d \ge \Delta(H)-g(k)$. This result is sharp up to the order of $g(k)$. %Subsequently, we investigate a concept called $\textit{path-pairability}$. A graph is said to be path-pairable if for any pairing of its vertices there exist a collection of edge-disjoint paths routing the the vertices of each pair. A question we are concerned here asks whether every planar path pairable graph on $n$ vertices must possess a vertex of degree linear in $n$. Indeed, we answer this question in the affirmative. We also sketch a proof resolving an analogous question for graphs embeddable on surfaces of bounded genus. Finally, in Chapter~$9$, we move on to examine $k$-linked tournaments. A tournament $T$ is said to be $k$-linked if for any two disjoint sets of vertices $\{x_1,\ldots ,x_k\}$ and $\{y_1,\dots,y_k\}$ there are directed vertex disjoint paths $P_1,\dots, P_k$ such that $P_i$ joins $x_i$ to $y_i$ for $i = 1,\ldots, k$. We prove that any $4k$ strongly-connected tournament with sufficiently large minimum out-degree is $k$-linked. This result comes close to proving a conjecture of Pokrovskiy