[Paper Review] An Improved Upper Bound for the Sum-free Subset Constant

Mark Lewko|arXiv (Cornell University)|Jul 16, 2010

Limits and Structures in Graph Theory3 references3 citations

TL;DR

This paper establishes a new upper bound of 11/28 ≈ 0.393 for the sum-free subset constant in finite sets of natural numbers, improving upon previous bounds. By constructing a specific 28-element set A and proving that no sum-free subset of size 12 exists within it, the authors demonstrate that the optimal constant in Erdős’s sum-free subset theorem cannot exceed 11/28, using a structured case analysis based on column-based doubling relations and sum constraints.

ABSTRACT

We show that the optimal constant in Erdös' sum-free subset theorem cannot be larger than $11/28 \approx .393$.

Motivation & Objective

• To improve the upper bound on the sum-free subset constant, which quantifies the largest possible fraction of elements in a finite set of natural numbers that can be sum-free.
• To resolve an open problem in extremal combinatorics concerning the sharpness of the 1/3 lower bound in Erdős’s sum-free subset theorem.
• To demonstrate that the optimal constant cannot exceed 11/28 by constructing a finite set where the largest sum-free subset is strictly smaller than 1/3 of the total size.
• To provide a human-verifiable proof that no sum-free subset of size 12 exists in a specific 28-element set A, thereby establishing the new upper bound.

Proposed method

• The authors define the sum-free subset constant δ(A) = l/|A|, where l is the size of the largest sum-free subset of a finite set A ⊂ ℕ.
• They use a tabular representation of the set A based on doubling relations (e.g., 1, 2, 4; 3, 6, 12; etc.) to structure the proof and constrain possible sum-free subsets.
• The proof proceeds by case analysis on the size of the intersection C = S ∩ B, where B is the union of the first three columns (elements 1–3, 4–6, 7–12), based on the number of elements from these columns in a hypothetical sum-free subset S of size 12.
• For each case (|C| = 2, 3, 4, 5), the authors derive exclusion constraints using sum relations (e.g., if x,y ∈ S then x+y ∉ S), and systematically eliminate all possible configurations.
• They use the fact that if d₁A ∩ d₂A = ∅, then the sum-free subset constant of d₁A ∪ d₂A is at most that of A, enabling the use of a single constructed set to bound the optimal constant.
• The proof is completed by showing that in all cases, assuming a sum-free subset of size 12 leads to a contradiction due to unavoidable sum relations.

Experimental results

Research questions

• RQ1 Can the optimal constant in Erdős’s sum-free subset theorem be strictly less than 1/3, and if so, what is the best possible upper bound?
• RQ2 What is the smallest possible sum-free subset constant across all finite sets of natural numbers, and can this be bounded below 11/28?
• RQ3 Is the set A = {1,2,3,...,18,20,22,24,25,26,27,30,34,50,54} with |A| = 28 the minimal example achieving a sum-free subset constant of 11/28?
• RQ4 Can a human-verifiable proof be constructed to show that no sum-free subset of size 12 exists in this specific set A?

Key findings

• The largest sum-free subset of the set A = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,20,22,24,25,26,27,30,34,50,54} contains exactly 11 elements.
• No sum-free subset of size 12 exists in A, as proven through a detailed case analysis on the intersection of such a subset with the first three columns of a structured tabular representation.
• The sum-free subset constant δ(A) = 11/28 ≈ 0.393 is achieved by this set, establishing it as a witness to the upper bound.
• The proof shows that any hypothetical sum-free subset of size 12 in A leads to a contradiction due to unavoidable sum relations, such as x + y = z with x,y,z ∈ S.
• The result improves upon previous upper bounds: 3/7 ≈ 0.429 (Erdős, 1965), 12/29 ≈ 0.414 (Alon and Kleitman, 1990), and 0.4 (Malouf, reported in 2002).
• The authors confirm that the bound 11/28 is tight for this construction and that no larger constant is possible, as the existence of such a set implies arbitrarily large sets with the same upper bound on the sum-free subset constant.