Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

Mittagsseminar (in cooperation with M. Ghaffari, A. Steger, D. Steurer and B. Sudakov)

Mittagsseminar Talk Information

Date and Time: Tuesday, September 17, 2019, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Miloš Trujić

Rödl-Ruciński theorem for large graphs

A seminal result of Rödl and Ruciński from 1995 states that for every fixed graph H a random graph G(n, p) is w.h.p. such that every 2-colouring of its edges contains a monochromatic copy of H in one of the two colours, provided that p >> n^(-1/m_2(H)), where the value m_2(H) is a certain (natural) density parameter of the graph H. We extend this theorem to all 'large' graphs H with maximum degree three. Namely, there is a b > 0 and a C > 0 for which w.h.p. a random graph G(n, p) is such that every 2-colouring of its edges contains a monochromatic copy of every graph H with b*n vertices and maximum degree at most three, provided that p >= Cn^(-2/5). The value 2/5 in the density p is optimal, which stems from the Rödl-Ruciński theorem. The statement has immediate consequences for the size-Ramsey numbers of large graphs, improving a result of Kohayakawa, Rödl, Schacht, and Szemerédi. This is a joint work with David Conlon and Rajko Nenadov.


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