Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, April 27, 2010, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Torsten Mütze

Globally sparse Ramsey Graphs

One of the fundamental topics in Ramsey theory is to study, for a forbidden graph F, the family of all graphs G with the property that any two-coloring of the edges of G contains a monochromatic copy of F (we say that such a graph G is F-Ramsey). The classical Ramsey number R(F) for instance is defined as the minimum number of vertices among all graphs G that are F-Ramsey (e.g., R(K_3)=6, as any graph on less than 6 vertices can be two-colored without a monochromatic triangle, and any two-coloring of the edges of K_6 contains a monochromatic triangle). Beside the minimum number of vertices among all F-Ramsey graphs, also the minimum number of edges, and the minimum clique-number among these graphs have been studied intensively. In this talk, we discuss results from [1] and [2] concerning the following question of this flavor: What is the minimum density among all graphs G that are F-Ramsey, where the density of a graph G is defined as m(G):=max_{H\subseteq G} e(H)/v(H) (the maximum is over all subgraphs H of G)? One of the results is, that the sparsest graph that is K_n-Ramsey is the complete graph on R(K_n) vertices.

Talk mainly based on
[1] A. Kurek and A. Rucinski, Globally sparse vertex-Ramsey graphs, J. Graph Theory 18 (1) (1994) 73--81.
[2] A. Kurek and A. Rucinski. Two variants of the size Ramsey number, Discuss. Math. Graph Th. 25 (2005) 141--149.

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