mathematics

Minimum overlap problem

Minimum overlap problem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Minimum overlap problem rather than just read about it. In short: In number theory and set theory, the minimum overlap problem is a problem proposed by Hungarian mathematician Paul Erdős in 1955. Formal statement of the problem Let A = {ai} and B = {bj} be two complementary subsets, a splitting of the set of natural numbers {1, 2, …, 2n}, such that both have the same cardinality, namely n.

Key takeaways

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

Reference excerpt

In number theory and set theory, the minimum overlap problem is a problem proposed by Hungarian mathematician Paul Erdős in 1955.

Formal statement of the problem Let A = {ai} and B = {bj} be two complementary subsets, a splitting of the set of natural numbers {1, 2, …, 2n}, such that both have the same cardinality, namely n. Denote by Mk the number of solutions of the equation ai − bj = k, where k is an integer varying between −2n and 2n. M (n) is defined as:

M ( n ) := min A , B max k M k . {\displaystyle M(n):=\min _{A,B}\max _{k}M_{k}.\,\!}

The problem is to estimate M (n) when n is sufficiently large.

History This problem can be found amongst the problems proposed by Paul Erdős in combinatorial number theory, known by English speakers as the Minimum overlap problem. It was first formulated in the 1955 article Some remarks on number theory (in Hebrew) in Riveon Lematematica, and has become one of the classical problems described by Richard K. Guy in his book Unsolved problems in number theory.

Partial results Since it was first formulated, there has been continuous progress made in the calculation of lower bounds and upper bounds of M (n), with the following results:

Lower

Upper

J. K. Haugland showed that the limit of M (n) / n exists and that it is less than 0.385694. For his research, he was awarded a prize in a young scientists competition in 1993. In 1996, he improved the upper bound to 0.38201 using a result of Peter Swinnerton-Dyer. This has now been further improved to 0.38093. In 2022, the lower bound was shown to be at least 0.379005 by E. P. White. In 2025, the AI system AlphaEvolve improved the upper bound to 0.380924, and in 2026 TTT-Discover, another AI system, further improved it to 0.380876. Later on, an open-sourced AI system, SimpleTES, has improved it to 0.380868.

The first known values of M(n) The values of M (n) for the first 15 positive integers are the following:

It is just the Law of Small Numbers that it is ⌊ 5 ( n + 3 ) / 13 ⌋ {\displaystyle \textstyle \lfloor 5(n+3)/13\rfloor }

References

Worked examples

Example 1 — a first encounter with Minimum overlap problem

Start with the simplest possible case. Write down what Minimum overlap problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Minimum overlap problem 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 Minimum overlap problem 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 Minimum overlap problem

In research: Minimum overlap problem appears in mathematics 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 Minimum overlap problem 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: Minimum overlap problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Additive number theory, Unsolved problems in number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Minimum overlap problem 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.

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How to study Minimum overlap problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Minimum overlap problem 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 Minimum overlap problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Minimum overlap problem in simple terms?

In number theory and set theory, the minimum overlap problem is a problem proposed by Hungarian mathematician Paul Erdős in 1955. Formal statement of the problem Let A = {ai} and B = {bj} be two complementary subsets, a splitting of the set of natural numbers {1, 2, …, 2n}, such that both have the…

Why does Minimum overlap problem matter?

Because it connects several mathematics 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 Minimum overlap problem?

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 Minimum overlap problem.