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Showing 1 to 9 of 9 for “"Ramsey numbers"”.
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Embedding Problems for Graphs and Hypergraphs
… I will also discuss graph and hypergraph Ramsey numbers, since two of the embedding results have important applications to Ramsey numbers which improve upon previously known results.
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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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Coloring Problems on Graphs and Hypergraphs
… this relates splittable colorings to classical Ramsey numbers. Let fr(m) be the least n such that some r-edge-coloring of K n is not (r, m)-splittable. Combinatorial designs yield fr(m) ≤ O(r2m2). Extending ideas of Erdo&huml;s and Gyarfas yields f r(m) ≥ min{O(rm 2), O(r2m)}. Similar …
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Games on Graphs and Other Combinatorial Problems
… 5, we consider the so-called restricted online Ramsey numbers, which correspond to a certain colouring game in the Builder-Painter setup. We provide a tight lower bound for the restricted online Ramsey numbers of matchings as long as the number of the allowed colours is small, resolving the …
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Results in Extremal Graph Theory, Ramsey Theory and Additive Combinatorics
… of Erdős and Simonovits on (ordinary) extremal numbers. In Chapter 5, we consider the following problem. Let 2 ≤ s < t be fixed integers. If G is an arbitrary K_t-free graph on n vertices, how large a K_s-free induced subgraph must there exist in G? This number, which is a generalisation of the …
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Extremal problems for labelling of graphs and distance in digraphs
… Furedi that appears in [26]. In Chapter 4 using Ramsey graphs, we determine the minimum clique size an n-vertex graph with chromatic number \chi can have if \chi \geq (n+3)/2. For integers n and t, we determine the maximum number of colors in an edge-coloring of a complete graph Kn that does not …
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Extremal Problems for Cycles, Paths and Set-Systems
… main tool (the Key Lemma) for proving results on Ramsey-type problems about cycles in sparse random graphs. In Chapter 3 we count at least how many Hamilton cycles one can find in a hypergraph which is guaranteed to contain at least one. For $0\leq \ell <k$, a Hamilton $\ell$-cycle in a …