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

Date and Time: Tuesday, November 22, 2011, 12:15 pm

Duration: 30 minutes

Location: OAT S15/S16/S17

Speaker: József Balogh (University of Illinois at Urbana-Champaign)

On the Ramsey-Turan numbers of graphs and hypergraphs

Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Turán number of H, RT_t(n, H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G with alpha_t(G) ≤ f(n), where alpha_t(G) is the maximum number of vertices in a K_t-free induced subgraph of G. Erdős, Hajnal, Simonovits, Sós, and Szemerédi posed several open questions about RT_t(n,K_s,o(n)), among them finding the minimum l such that RT_t(n,K_t+l,o(n)) = Ω(n^2). We answer this question by proving that RT_t(n,K_t+2,o(n)) = Ω(n^2). From the other side, it is easy to see that RT_t(n,K_t+1,o(n)) = o(n^2). Our constructions which show that RT_t(n,K_t+2,o(n)) = Ω(n^2) also imply several results on the Ramsey-Turán numbers of hypergraphs.

It is joint work with John Lenz.

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