[Paper Review] Random sum-free subsets of Abelian groups

József Balogh, Robert Morris+1 more|arXiv (Cornell University)|Mar 10, 2011

Limits and Structures in Graph Theory20 references4 citations

TL;DR

This paper characterizes the structure of maximum-size sum-free subsets in random subsets of Abelian groups, showing that for groups $ G $ with $ |G| $ divisible by a prime $ q \equiv 2 \pmod{3} $, such subsets are typically contained in maximum sum-free subsets of $ G $ when the random subset is dense enough. For $ \mathbb{Z}_{2n} $, it establishes a sharp threshold for this property using transference theorems and stability results in additive combinatorics.

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 $p_c \approx \sqrt{\log n / n}$ above which, with high probability as $|G| o \infty$, each such subset is contained in a maximum-size sum-free subset of $G$, whenever $q$ divides $|G|$ for some (fixed) prime $q$ with $q \equiv 2 \pmod 3$. Moreover, in the special case $G = \ZZ_{2n}$, we determine a 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.

Motivation & Objective

• To determine the threshold density above which, with high probability, maximum-size sum-free subsets of a random subset of an Abelian group $ G $ are contained in some maximum-size sum-free subset of $ G $.
• To establish the existence and exact value of the threshold function for this structural containment property in random sum-free subsets of $ G $, particularly when $ |G| $ is divisible by a prime $ q \equiv 2 \pmod{3} $.
• To prove a sharp threshold result specifically for the group $ \mathbb{Z}_{2n} $, where the transition from non-containment to containment occurs precisely at a critical density.
• To extend the understanding of random analogues of extremal additive combinatorics problems by combining transference principles with stability theorems for sum-free sets.
• To resolve a long-standing open problem on the structure of largest sum-free subsets in random subsets of $ \mathbb{Z}_n $, building on recent advances in random combinatorics.

Proposed method

• Utilizes recent transference theorems by Conlon and Gowers to lift results from dense sets to random sparse sets in Abelian groups.
• Applies stability theorems for sum-free subsets in Abelian groups, particularly those of Green and Ruzsa, to analyze the structure of extremal sum-free sets.
• Employs Janson's inequality to bound the probability that a random graph associated with a candidate sum-free set has no edges, controlling the expected number of exceptional configurations.
• Introduces a refined counting argument over tuples $ (S, T, U) $ representing potential sum-free configurations, bounding the expected number of such configurations via exponential moment estimates.
• Uses probabilistic method with moment bounds and tail estimates, distinguishing cases based on the relative size of $ \mu' $ and $ \Delta' $ to control the expectation of the number of non-structured sum-free sets.
• Applies asymptotic estimates on binomial coefficients and logarithmic bounds to control the growth of the number of configurations, ensuring the total expected number of bad configurations tends to zero.

Experimental results

Research questions

• RQ1 For which densities $ p $ is it true, with high probability, that every maximum-size sum-free subset of a $ p $-random subset of an Abelian group $ G $ is contained in some maximum-size sum-free subset of $ G $, when $ q \mid |G| $ and $ q \equiv 2 \pmod{3} $?
• RQ2 What is the exact threshold function $ p_c(n) $ for the structural containment property in random sum-free subsets of $ \mathbb{Z}_{2n} $, and is it sharp?
• RQ3 How do transference theorems from dense to random settings apply to the problem of sum-free sets in Abelian groups, and what structural constraints emerge?
• RQ4 Can the expected number of 'bad' sum-free configurations—those not contained in any maximal sum-free set—be shown to vanish as $ n \to \infty $, and under what conditions?
• RQ5 What role do stability theorems for sum-free sets play in proving that random sum-free sets inherit the structure of their ambient group’s extremal sum-free sets?

Key findings

• For Abelian groups $ G $ with $ |G| $ divisible by a prime $ q \equiv 2 \pmod{3} $, there exists a threshold $ p_c $ such that for $ p \gg p_c $, with high probability, every maximum-size sum-free subset of a $ p $-random subset of $ G $ is contained in some maximum-size sum-free subset of $ G $.
• In the special case $ G = \mathbb{Z}_{2n} $, the threshold for the above structural containment property is sharp, meaning the transition from non-containment to containment occurs precisely at a critical density.
• The threshold function for $ \mathbb{Z}_{2n} $ is shown to be sharp, with the property holding for $ p > (1+\varepsilon)p_c $ and failing for $ p < (1-\varepsilon)p_c $, for any fixed $ \varepsilon > 0 $.
• The expected number of sum-free subsets of size $ k $ that are not contained in any maximal sum-free set of $ G $ decays as $ n^{-\varepsilon k} $ or faster, ensuring such configurations vanish asymptotically.
• The proof relies on bounding the expectation of the number of 'bad' configurations using Janson's inequality and moment estimates, showing the total expectation tends to zero as $ n \to \infty $.
• The authors establish that for $ p \gg n^{-1/2} $, the structure of the largest sum-free subsets in random subsets of $ \mathbb{Z}_{2n} $ is asymptotically the same as in the full group, confirming a conjectured structural stability.