- Bedert solved a 60-year-old Erdős conjecture about sum-free sets.
- Any set of N integers has a sum-free subset larger than N/3.
- The proof bridges structured and random sets using Fourier analysis.
Benjamin Bedert was reading a 1997 paper by the late Fields Medalist Jean Bourgain when something clicked. The Oxford graduate student had spent months circling a conjecture about sum-free sets, one of the oldest unsolved problems in additive combinatorics.
Over Christmas 2024, the pieces fell into place.
The result, posted to arXiv in February 2025, settled a question that Paul Erdős had posed six decades earlier. It also revealed something unexpected about the hidden structure of ordinary numbers.
A Deceptively Simple Question About Addition
Sum-free sets sound almost trivial. Take any collection of numbers. Can you find a subset where no two members add up to a third?
The odd numbers qualify: add any two odd numbers and you get an even one, safely outside the set.
Sum-free sets
A sum-free set is a collection of numbers where no two elements add up to another element in the set. The odd numbers are sum-free because any two odd numbers always sum to an even number. The question is how large such subsets can be within any given set of integers.
Erdős proved in 1965 that for any set of N integers, you can always extract a sum-free subset containing at least N/3 elements. A clean result.
But he suspected the true answer was larger, and said so with characteristic understatement: "It is surprising that this simple question seems to present considerable difficulties."
He was right about the difficulties. For thirty years, progress barely moved. Two researchers nudged the bound to (N+1)/3 in 1990.
Bourgain, with typical ingenuity, pushed it to (N+2)/3 in 1997 and sketched a possible path forward using a tool from Fourier analysis called the Littlewood norm.
Then the problem went quiet.
Fourier Analysis Cracks the Combinatorics
Bedert, working under number theorist Ben Green at Oxford, picked up Bourgain's trail in the summer of 2024. The Littlewood norm measures something intuitive: how structured or random a set of numbers appears when you decompose it into frequency components.
Key figure
N/3 + c log(log N)
Bedert's new lower bound for the size of sum-free subsets in any set of N integers, confirming Erdős's conjecture that N/3 was not the best possible.
His insight was that sets with a small Littlewood norm, the structured ones, behave like arithmetic progressions. Evenly spaced sequences contain many pairs that produce identical sums, which makes finding sum-free subsets easier.
For sets with a large Littlewood norm, Bourgain's existing techniques already applied.
The gap between these two cases had stalled everyone. Bedert bridged it by modifying a 1981 proof to show that any set's components must have a large Littlewood norm, then used Bourgain's methods to finish the argument.
The result: any set of N integers contains a sum-free subset of at least N/3 + c log(log N) elements. The numerical improvement is modest. For a set of 10100 numbers, the gain is roughly 5 extra elements.
But the correction grows without limit as N increases. That is precisely what Erdős had conjectured.
Old and difficult problem solved by brilliant kid.
Julian Sahasrabudhe, University of Cambridge
A Massive Gap Remains
Julian Sahasrabudhe, a combinatorialist at Cambridge who now supervises Bedert's postdoctoral research, called the problem "a very basic-sounding thing that we had shockingly little understanding of."
His summary was more concise: "Old and difficult problem solved by brilliant kid."
Ben Green, Bedert's doctoral adviser at Oxford, noted that Erdős's bound had resisted improvement so stubbornly that any progress at all was remarkable. "It's very, very hard to do any better at all," Green said.
Yet the problem is far from closed. The deviation beyond N/3 grows as log(log N) in Bedert's proof, but the theoretical upper bound allows growth as fast as N itself.
Green describes this as "a massive gap" between what is proven and what might be true.
Where the Numbers Lead Next
Bedert completed his DPhil at Oxford in August 2025 and moved to Cambridge to work with Sahasrabudhe. Sean Eberhard, a mathematician at the University of Warwick, captured the appeal of the remaining puzzle.
"It's beautiful, it's interesting, it feels natural," Eberhard said. "You want to solve a mystery, don't you?"
Bedert himself offered a quieter reflection on how the solution emerged. "I'm not sure where it came from," he said. "Maybe these ideas stir in your mind for a while, and then finally you get something out that works."
The sum-free set problem waited sixty years for that moment of stirring to resolve into proof.
It joins a tradition of legendary math problems where persistence eventually wins.
Sources
- Primary Research: Large sum-free subsets of sets of integers via L¹-estimates for trigonometric series (Benjamin Bedert, 2025)
- Additional Context:
- Graduate Student Solves Classic Problem About the Limits of Addition (Quanta Magazine)
- Benjamin Bedert Faculty Page (University of Cambridge)
