Mantel's theorem for random graphs

For a graph G, denote by tG resp. bG the maximum size of a triangle-free resp. bipartite subgraph of G. Of course tGi¾?bG for any G, and a classic result of Mantel from 1907 the first case of Turan's Theorem says that equality holds for complete graphs. A natural question, first considered by Babai, Simonovits and Spencer about 20 years ago is, when i.e., for what p=pn is the "Erdi¾?s-Renyi" random graph G=Gn,p likely to satisfy tG=bG? We show that this is true if p>Cn-1/2logi¾?1/2n for a suitable constant C, which is best possible up to the value of C. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 59-72, 2015

[1]  Svante Janson,et al.  Random graphs , 2000, Wiley-Interscience series in discrete mathematics and optimization.

[2]  Roy Meshulam,et al.  The Clique Complex and Hypergraph Matching , 2001, Comb..

[3]  Wojciech Samotij.  Stability results for random discrete structures , 2014, Random Struct. Algorithms.

[4]  Vojtech Rödl,et al.  Large triangle-free subgraphs in graphs withoutK4 , 1986, Graphs Comb..

[5]  J. Balogh,et al.  Independent sets in hypergraphs , 2012, 1204.6530.

[6]  Béla Bollobás,et al.  Random Graphs , 1985 .

[7]  A. Goodman.  On Sets of Acquaintances and Strangers at any Party , 1959 .

[8]  Y. Kohayakawa.  Szemerédi's regularity lemma for sparse graphs , 1997 .

[9]  J. Kahn,et al.  On the triangle space of a random graph , 2012, 1207.6717.

[10]  V. Rödl,et al.  Arithmetic progressions of length three in subsets of a random set , 1996 .

[11]  T. Lu.  ON K4-FREE SUBGRAPHS OF RANDOM GRAPHS , 1997 .

[12]  E. Szemerédi.  Regular Partitions of Graphs , 1975 .

[13]  Alan M. Frieze,et al.  Random graphs , 2006, SODA '06.

[14]  Svante Janson,et al.  Random graphs , 2000, ZOR Methods Model. Oper. Res..

[15]  Konstantinos Panagiotou,et al.  Extremal subgraphs of random graphs , 2007, SODA '07.

[16]  W. T. Gowers,et al.  Combinatorial theorems in sparse random sets , 2010, 1011.4310.

[17]  Vojtech Rödl,et al.  The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent , 1986, Graphs Comb..

[18]  M. Schacht.  Extremal results for random discrete structures , 2016, 1603.00894.

[19]  Matthew Kahle,et al.  Topology of random clique complexes , 2006, Discret. Math..