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Showing 1 to 11 of 11 for “"Ramsey number"”.
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On vertex degrees, graph decomposition, and circular chromatic Ramsey number
… problems about vertex degrees and a variant of Ramsey number of graphs, and also structural problems about graph decomposition. In a list (d_1,...,d_n) of positive integers, let r and s denote the largest and smallest entries. A list is gap-free if each integer between r and s is present. In …
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Ramsey Theory
<p>The Ramsey number $R(r, b)$ is the least positive integer such that every edge 2-coloring of the complete graph $K_{R(r, b)}$ with colors red and blue either embeds a red $K_r$ or a blue $K_b$. We explore various methods to find lower bounds on $R(r,b)$, finding new results on fibrations and …
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Extremal Problems for Partitions of Edge Sets of Graphs
… The first family we address consists of induced Ramsey number problems. Induced Ramsey numbers generalize ordinary Ramsey numbers. The induced Ramsey number of a pair of graphs G and H is the smallest n such that there exists a graph F on n vertices such that any partition of the edge set of F …
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Cliques in graphs
… is to evaluate $k_r(n,\delta)$, the minimal number of $r$-cliques in graphs with $n$ vertices and minimum degree~$\delta$. A fundamental result in Graph Theory states that a triangle-free graph of order $n$ has at most $n^2/4$ edges. Hence, a triangle-free graph has minimum degree at most …
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Results in Ramsey theory and extremal graph theory
… relating to graphs. The first problem is in Ramsey theory, while the others are in extremal graph theory. In Chapter 2, which is joint work with Vojtěch Dvořák, we consider the Ramsey number $R(F_n)$ of the fan graph $F_n$, a graph consisting of $n$ triangles which all share a common vertex. …
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Ramsey theory: The Erdős-Gyárfás problem and ordered size Ramsey questions
DSpace SAF Submission Ingestion Package generated from Vireo submission #16387 on 2021-09-16 at 16:42:26
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Topics in Probabilistic Combinatorics
… graphs. In Chapter 5, we investigate the online Ramsey number of paths, showing that for every $k\ge 10$, the online Ramsey number for paths $P_k$ and $P_n$ satisfies $\tilde{r}(P_k,P_n) \geq \frac{5}{3}n + \frac{k}{9} - 4$. This matches up to a linear term in $k$ the upper bound recently …
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Extremal Problems for Cycles, Paths and Set-Systems
… interest. In Chapter 2 we study the Turán number of long cycles in random and pseudo-random graphs. Denote by $ex(G(n,p),H)$ the random variable counting the number of edges in a largest subgraph of $G(n,p)$ without a copy of $H$. We determine the asymptotic value of $ex(G(n,p), C_t)$ where …
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Extremal problems on special graph colorings
Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-10-02 without embargo terms