2012/01/31 by Alon, Noga, Balogh, József, Morris, Robert +1
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1201.6654
In this paper we study sum-free sets of order m in finite Abelian groups. We prove a general theorem on 3-uniform hypergraphs, which allows us to deduce structural results in the sparse setting from stability results in the dense setting. As a consequence, we determine the typical structure and asymptotic number of sum-free sets of order m in Abelian groups G whose order is divisible by a prime q with q ≡ 2 \pmod 3, for every m ≥ C(q) √(n log n), thus extending and refining a theorem of Green and Ruzsa. In particular, we prove that almost all sum-free subsets of size m are contained in a maximum-size sum-free subset of G. We also give a completely self-contained proof of this statement for Abelian groups of even order, which uses spectral methods and a new bound on the number of independent sets of size m in an (n,d,λ)-graph.