Ponencia
Phase transitions in the Ramsey-Turán theory
Autor/es | Balogh, József |
Coordinador/Director | Díaz Báñez, José Miguel
Garijo Royo, Delia Márquez Pérez, Alberto Urrutia Galicia, Jorge |
Departamento | Universidad de Sevilla. Departamento de Matemática Aplicada II |
Fecha de publicación | 2013 |
Fecha de depósito | 2017-06-07 |
Publicado en |
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Resumen | Let f(n) be a function and L be a graph. Denote by RT(n, L, f(n)) the maximum number of edges of an L-free graph on n vertices with independence number less than f(n). Erdos and Sós asked if RT (n, K5, c√
n) = o (n2) for ... Let f(n) be a function and L be a graph. Denote by RT(n, L, f(n)) the maximum number of edges of an L-free graph on n vertices with independence number less than f(n). Erdos and Sós asked if RT (n, K5, c√ n) = o (n2) for some constant c. We answer this question by proving the stronger RT(n, K5, o (√n log n)) = o(n2). It is known that RT (n, K5, c√n log n )= n2/4 + o (n2) for c > 1, so one can say that K5 has a Ramsey-Turán-phase transition at c√n log n. We extend this result to several other Kp's and functions f(n), determining many more phase transitions. We shall formulate several open problems, in particular, whether variants of the Bollobás-Erdos graph, which is a geometric construction, exist to give good lower bounds on RT (n, Kp, f(n)) for various pairs of p and f(n). These problems are studied in depth by Balogh-HuSimonovits, where among others, the Szemerédi's Regularity Lemma and the Hypergraph Dependent Random Choice Lemma are used. |
Identificador del proyecto | DMS-0745185 |
Cita | Balogh, J. (2013). Phase transitions in the Ramsey-Turán theory. En XV Spanish Meeting on Computational Geometry, Sevilla. |
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