Random sum-free subsets of abelian groups

Abstract

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 = Z 2n , 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 subsets of Abelian groups.

Stability results for random discrete structures

“…First, note that p 3 n |H n | = Ω(n 1/2 ). Since for each Schur triple {x, y, z} ∈ H n , there are only constantly many Schur triples {x ′ , y ′ , z ′ } ∈ H n intersecting {x, y, z} in more than one element, an easy computation (see [21]) shows that H is (K, p)-bounded for sufficiently large constant K. Finally, since |B n | ≤ n (see, e.g., [3,Corollary 3.4]), we have that |B n | = exp(o(n 1/2 )) = exp(o(p n |V (H n )|). Crucially, we need to verify that H is 1 3 + 1 3q , B -stable.…”

“…As a last application of our main result, we will give a much more transparent proof of the following sparse random analogue of Theorem 1.10, originally derived from the transference theorem of Conlon and Gowers [4] by Balogh, Morris, and Samotij [3], with an improved probability estimate. Theorem 1.11 ([3, 4]).…”

“…Since throughout the paper, we will deal with many sequences indexed by (subsets of) the natural numbers, in order to unclutter the notation and, hopefully, improve readability, we use the (somewhat informal) notational convention that the sequences are denoted by boldface letters, e.g., p stands for (p n ), that is, the sequence p : N → [0, 1] indexed by the set of natural numbers 3 . the only exception is that, due to typesetting limitations, the sequence (B n ) will be denoted by B.…”

“…Similarly as in Theorem 1.3, the methods of Conlon and Gowers allow one to prove the above statement only in the case when H is strictly -balanced, 3 i.e., if it has the largest -density among all of its subhypergraphs or, in other words, if every proper subhypergraph H H satisfies m (H ) < m (H).…”

“…. , 7} with edge set {1234, 1235, 4567}), proved by Pikhurko [23]; the hypergraphs C (2k) 3 (the 2k-uniform hypergraphs on the vertex set {1, . .…”