TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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Every nonempty finite set of $n$ nonzero integers has a sum-free subset of size greater than $n / 3$

Statement

Every nonempty finite set $A$ of $n$ nonzero integers contains a sum-free subset of cardinality strictly greater than $n / 3$ .

Facts & Assumptions

Given: A nonempty finite set $A \subseteq Z ∖ {0}$ with $∣ A ∣ = n$ .

[L1]

Sum-free means no three members, with repeated summands allowed, satisfy $x + y = z$ (Sum-free subsets of the integers).

[L2]

There are primes $p$ arbitrarily large with $p \equiv 2 (m o d 3)$ (There are arbitrarily large primes congruent to $2$ modulo $3$ ).

[L3]

For prime $p$ , nonzero residue classes form the multiplicative group of the field $Z / p$ (For every prime $p$ , the two operations on $Z / p$ make it a field).

[L4]

A nonempty finite set of real numbers has a maximum (Every nonempty finite set of reals has a maximum and a minimum).

[L5]

Uniform probabilities are cardinality ratios; indicators count membership, expectation is linear, and some outcome reaches at least the expectation (The uniform probability space on a nonempty finite set, Indicators turn event probabilities, intersections, and finite counts into expectations and products, Expectation is linear for every finite family of random variables, without any independence hypothesis, Expectation preserves pointwise order and lies between the minimum and maximum attained values).

Proof

technique · constructive

1.1

Since $A$ is nonempty, $n \geq 1$ . Choose a prime $p = 3 r + 2$ larger than twice $\max_{a \in A} ∣ a ∣$ .

givenL2L4chooseconstruct

1.2

Let $I = {r + 1, \dots, 2 r + 1} \subset Z / p$ . This set is sum-free modulo $p$ : if representatives $x, y, u \in I$ satisfied $x + y \equiv u (m o d p)$ , then $x + y - u$ would be a multiple of $p$ lying between $1$ and $3 r + 1 = p - 1$ , which is impossible.

L1algebra

2.1

Choose $z$ uniformly from the nonzero residue classes and put $B_{z} = {a \in A : z a m o d p \in I}$ . Reduction modulo $p$ is injective on $A$ by the choice of $p$ , and multiplication by each nonzero $a$ permutes the nonzero classes by [L3].

step 1.1L3

3.1

Thus every $a \in A$ belongs to $B_{z}$ with probability $∣ I ∣ / (p - 1) = (r + 1) / (3 r + 1) > 1 / 3$ . Linearity gives $E [∣ B_{z} ∣] = n (r + 1) / (3 r + 1) > n / 3$ .

step 2.1L5algebra

4.1

Some $z$ has $∣ B_{z} ∣$ at least the expectation. If $x + y = u$ in $B_{z}$ , then $z x + z y = z u$ modulo $p$ , contradicting sum-freeness of $I$ ; so $B_{z}$ is sum-free and has size greater than $n / 3$ .

step 1.2step 3.1L1L5discharge-construct∎

Depends on

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Sources

M. Bucic, Probabilistic Method, Theorem 2.3 (standard reference, not scraped)

Every nonempty finite set of nn nonzero integers has a sum-free subset of size greater than n/3n/3