Random sum-free subsets of abelian groups

We characterize the structure of maximum-size sum-free subsets of a random subset of an abelian group G. In particular, we determine the threshold above which, with high probability as |G| → ∞, each such subset is contained in some maximum-size sum-free subset of G, whenever q divides |G| for some (fixed) prime q with q ≡ 2 (mod 3). Moreover, in the special case G = ℤ2n, we determine the sharp threshold for the above property. The proof uses recent ‘transference’ theorems of Conlon and Gowers, together with stability theorems for sum-free sets of abelian groups.

[1]  T. Luczak.  Randomness and regularity , 2006 .

[2]  E. Friedgut,et al.  Sharp thresholds of graph properties, and the -sat problem , 1999 .

[3]  Gregory A. Freiman.  On the structure and the number of sum-free sets , 1992 .

[4]  I. Schur.  Über Kongruenz x ... (mod. p.). , 1917 .

[5]  Andrzej Ruciński,et al.  Rado Partition Theorem for Random Subsets of Integers , 1997 .

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

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

[8]  Neil J. Calkin.  On the Number of Sum-Free Sets , 1990 .

[9]  Paul Erdös,et al.  Notes on Sum-Free and Related Sets , 1999 .

[10]  Vsevolod F. Lev,et al.  Cameron-Erdo? s Modulo a Prime , 2002 .

[11]  H. P. Yap,et al.  Maximal Sum-Free Sets of Elements of Finite Groups , 1969 .

[12]  Cameron-Erdős Modulo a Prime , 2002 .

[13]  Vojtech Rödl,et al.  A sharp threshold for random graphs with a monochromatic triangle in every edge coloring , 2006, Memoirs of the American Mathematical Society.

[14]  Vojtech Rödl,et al.  Quantitative theorems for regular systems of equations , 1988, J. Comb. Theory, Ser. A.

[15]  V. Rödl,et al.  Ramsey properties of random discrete structures , 2010 .

[16]  H. Abbott,et al.  Sum-free sets of integers , 1966 .

[17]  Miklós Simonovits,et al.  Extremal subgraphs of random graphs , 1990, J. Graph Theory.

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

[19]  Vojtech Rödl,et al.  On Schur Properties of Random Subsets of Integers , 1996 .

[20]  B. Green.  A Szemerédi-type regularity lemma in abelian groups, with applications , 2003, math/0310476.

[21]  G. Kalai,et al.  Every monotone graph property has a sharp threshold , 1996 .

[22]  Maximal Sum-Free Sets of Group Elements , 1969 .

[23]  alcun K. grafo.  ASYMPTOTIC ENUMERATION OF Kn-FREE GRAPHS , 2004 .

[24]  V. Rödl,et al.  Threshold functions for Ramsey properties , 1995 .

[25]  N. Alon,et al.  The Probabilistic Method: Alon/Probabilistic , 2008 .

[26]  A. A. Sapozhenko.  The Cameron-Erd˝ os conjecture , 2008 .

[27]  N. Alon,et al.  A Tribute to Paul Erdős: Sum-free subsets , 1990 .

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

[29]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[30]  K. F. Roth.  On Certain Sets of Integers , 1953 .

[31]  Hamed Hatami.  A structure theorem for Boolean functions with small total influences , 2010, 1008.1021.

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

[33]  N. Alon.  Independent sets in regular graphs and sum-free subsets of finite groups , 1991 .

[34]  Béla Bollobás,et al.  Threshold functions , 1987, Comb..

[35]  Peter J. Cameron.  On the structure of a random sum-free set , 1987 .

[36]  Daniel Král,et al.  A combinatorial proof of the Removal Lemma for Groups , 2008, J. Comb. Theory, Ser. A.

[37]  Wojciech Samotij,et al.  Counting sum-free sets in abelian groups , 2012, 1201.6654.

[38]  Ben Green,et al.  Sum-free sets in abelian groups , 2003 .

[39]  Ben Green,et al.  Counting sumsets and sum-free sets modulo a prime , 2004 .

[40]  On sum-free subsequences , 1975 .

[41]  Richard Mollin,et al.  On the Number of Sets of Integers With Various Properties , 1990 .

[42]  Ben Green.  The Cameron–Erdős Conjecture , 2003 .