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‘Huge Breakthrough’ in the Math of Imbalance
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combinatorics
‘Huge Breakthrough’ in the Math of Imbalance
By
Max G. Levy
August 21, 2026
For the first time in 30 years, computer scientists have found a better way to allocate objects evenly between two groups.
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Ada Zejun Shen for Quanta Magazine
By Max G. Levy
Contributing Writer
August 21, 2026
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algorithms
combinatorics
computer science
mathematics
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One does not need a doctorate in mathematics to split 12 eager trivia buffs into two competitive teams. But consider that each person arrives with unique strengths and liabilities: One may be a geography obsessive with no ear for music, another could be a naturalist who doesn’t own a television, and another could be a cinephile who never reads. Balancing traits between two camps becomes a lot harder.
So, how evenly can you assemble the teams so that they have matching firepower in every category, from Greek mythology to college basketball?
You can always make the teams surprisingly even, according to researchers studying combinatorial discrepancy theory.
Discrepancy theory is a branch of mathematics concerned with allocating resources as evenly as possible. If one trivia team gets all the history knowledge, leaving none for the other, that’s a big discrepancy.
In the early 1980s, the mathematician János Komlós came up with a counterintuitive prediction. He conjectured that no matter how many objects (your players) or dimensions (trivia categories) you consider, the discrepancy — which you can quantify — will never exceed a constant amount. There will always be a way to divide the teams with a discrepancy below that exact amount.
“This is really astonishing,” said Haotian Jiang, a theoretical computer scientist at the University of Chicago. “The Komlós conjecture says it has nothing to do with the dimension of the problem. It’s a universal constant.”
No one has ever found a way to contradict the conjecture. Yet it is so astonishing that some mathematicians thought it must be false. Proving it is “one of these holy-grail problems in discrepancy theory,” said Nikhil Bansal, a theoretical computer scientist from the University of Michigan.
Even the conjecture’s creator thinks it’s somewhat absurd. “I was young and foolish when I made it,” the now retired Komlós joked in an email. “I threw a wrench into combinatorial discrepancy theory with this irresponsible conjecture.”
If the Komlós conjecture is true, it could unlock answers to many other problems, both within discrepancy theory and in fields like operations research.
But for decades, a proof looked like a long shot. Mathematicians weren’t able to make much progress; their best upper limit on the discrepancy, achieved in 1998, still depended strongly on the dimension of the problem. It was far from constant.
Then, in fall 2025, Bansal and Jiang announced the first major advance on the problem in nearly 30 years. They found a limit that changes so slowly with the dimension that it is only a hair away from constant, even with an astronomical number of dimensions. Other researchers described the work, which used a novel algorithmic approach, as “very exciting,” “a beautiful result,” and “a huge step forward.”
While the unexpected finding has not fully resolved the problem, it offers the most compelling evidence yet that Komlós’ conjecture wasn’t so irresponsible after all. “I used to lean toward thinking the conjecture is false,” said Aleksandar Nikolov, a computer scientist at the University of Toronto. The new work “is now making me quite a bit more confident that probably the conjecture actually is true.”
Bansal and Jiang’s solution shows how unfathomably complex systems can be wrangled into something much simpler and easier to study — and offers insights that have potential applications in math, physics, and even machine learning.
Divide and Conquer
Discrepancy problems like Komlós’ deal with breaking sets of objects into two subsets. You can think of splitting people into trivia teams, or used cars into lots, or clinical trial participants into treatment and placebo groups.
The Komlós conjecture imagines each person (or object) as an arrow of length 1 called a unit vector. This vector is defined by a list of coordinates, where each coordinate measures how much of a particular attribute that person has.
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