Sum-free set

Key takeaways

Sum-free set belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Sum-free set to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Sum-free set from memory before moving on to harder problems.

Reference excerpt

In additive combinatorics and number theory, a subset A of an abelian group G is said to be sum-free if the sumset A + A is disjoint from A. In other words, A is sum-free if the equation a + b = c {\displaystyle a+b=c} has no solution with a , b , c ∈ A {\displaystyle a,b,c\in A} . For example, the set of odd numbers is a sum-free subset of the integers, and the set {N + 1, ..., 2N} forms a large sum-free subset of the set {1, ..., 2N}. Fermat's Last Theorem is the statement that, for a given integer n > 2, the set of all nonzero nth powers of the integers is a sum-free set. Some basic questions that have been asked about sum-free sets are:

How many sum-free subsets of {1, ..., N} are there, for an integer N? Ben Green has shown that the answer is O ( 2 N / 2 ) {\displaystyle O(2^{N/2})} , as predicted by the Cameron–Erdős conjecture. How many sum-free sets does an abelian group G contain? What is the size of the largest sum-free set that an abelian group G contains? A sum-free set is said to be maximal if it is not a proper subset of another sum-free set. Let f : [ 1 , ∞ ) → [ 1 , ∞ ) {\displaystyle f:[1,\infty )\to [1,\infty )} be defined by f ( n ) {\displaystyle f(n)} is the largest number k {\displaystyle k} such that any set of n nonzero integers has a sum-free subset of size k. The function is subadditive, and by the Fekete subadditivity lemma, lim n f ( n ) n {\displaystyle \lim _{n}{\frac {f(n)}{n}}} exists. Erdős proved that lim n f ( n ) n ≥ 1 3 {\displaystyle \lim _{n}{\frac {f(n)}{n}}\geq {\frac {1}{3}}} , and conjectured that equality holds. This was proved in 2014 by Eberhard, Green, and Manners giving an upper bound matching Erdős' lower bound up to a function or order o ( n ) {\displaystyle o(n)} , f ( n ) ≤ n 3 + o ( n ) {\displaystyle f(n)\leq {\frac {n}{3}}+o(n)} . Erdős also asked if f ( n ) ≥ n 3 + ω ( n ) {\displaystyle f(n)\geq {\frac {n}{3}}+\omega (n)} for some ω ( n ) → ∞ {\displaystyle \omega (n)\rightarrow \infty } , in 2025 Bedert in a preprint proved this giving the lower bound f ( n ) ≥ n 3 + c log ⁡ log ⁡ n {\displaystyle f(n)\geq {\frac {n}{3}}+c\log \log n} .

See also Erdős–Szemerédi theorem Sum-free sequence

References

External links Erdős Problem 792 - erdosproblems.com

Worked examples

Example 1 — a first encounter with Sum-free set

Start with the simplest possible case. Write down what Sum-free set claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Sum-free set before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Sum-free set ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Sum-free set

In research: Sum-free set appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Sum-free set in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Sum-free set is common in secondary-school and first-year university syllabi. It links to neighbouring topics Additive combinatorics, Sumsets, so understanding it makes those chapters shorter.
In everyday life: Look for Sum-free set outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Sum-free set in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Sum-free set means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Sum-free set out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Sum-free set in simple terms?

In additive combinatorics and number theory, a subset A of an abelian group G is said to be sum-free if the sumset A + A is disjoint from A. In other words, A is sum-free if the equation a + b = c {\displaystyle a+b=c} has no solution with a , b , c ∈ A {\displaystyle a,b,c\in A} .

Why does Sum-free set matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Sum-free set?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Sum-free set.