Scientists Disprove Bunkbed Conjecture

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Mathematicians from Russia, including two HSE graduates, have disproven a well-known mathematical conjecture that, despite lacking solid proof, had been considered valid for 40 years. The ‘Bunkbed Conjecture’ belongs to percolation theory—a branch of mathematics that studies the formation of connected structures in independent environments.

The hypothesis was proposed in the 1980s by Dutch physicist Pieter Kasteleyn, who aimed to mathematically describe how liquids seep through porous surfaces, such as water saturating a sponge.

The conjecture is based on random connections between vertices in an imaginary graph resembling a bunkbed. It posits that the probability of a connection forming between two vertices on the same level is higher than the probability of a connection forming between levels.

© Mark Belan / Quanta Magazine

While this statement seems intuitively true, no convincing proof had been found to confirm or refute it until recently. Sceptics argued that the claim was too general to hold true in all cases.

Mathematics typically focuses on proving the validity of statements, with disproofs being relatively rare. However, a team of Russian mathematicians—Igor Pak, Nikita Gladkov, and Aleksandr Zimin—managed to find a counterexample that invalidated the conjecture.

‘Actually, my colleague Nikita Gladkov and I first encountered the “bunkbed” concept during our freshman year at HSE. We were dorm roommates, and our room actually had a bunkbed,’ joked Aleksandr Zimin. ‘With this conjecture, we understood that it holds true for most cases. But we were curious—are there rare cases where it fails?’

Initially, the team tried to find a counterexample using machine learning methods. They trained a neural network to identify potential connections in graphs and attempted to explore all possible configurations. However, for graphs with more than nine vertices, the number of possible connections grew exponentially, quickly exceeding computational limits. No proof was found.

The researchers then adapted methods from hypergraph theory, where a disproof of the Bunkbed Conjecture already existed, to classical graphs. They constructed a highly complex structure containing thousands of vertices and edges. In this graph, the probability of a connection forming between the upper and lower levels was slightly higher than the probability of a connection forming on the lower level, thereby disproving the conjecture.

Picture 1. Hypergraph used as the basis for the disproof

© The Bunkbed Conjecture is False / Nikita Gladkov, Igor Pak, Aleksandr Zimin / arXiv:2410.02545v1 [math.CO]

‘My co-authors Igor Pak and Nikita Gladkov—who are currently working at UCLA—and I complemented each other perfectly on this project. I prefer using numerical methods. In my opinion, to truly understand a problem, you need to be able to program it and explain it to a computer. Nikita, on the other hand, takes a different approach and prefers relying on a more abstract, intuitive level,’ says Aleksandr Zimin, one of the authors of the paper and a postgraduate student at the HSE Faculty of Mathematics.

Aleksandr Zimin

‘The conjecture resisted disproof for a long time—or perhaps no one wanted to disprove it because it was beautiful and elegant. However, in my view, disproving it does not destroy its beauty; rather, it proves that the world is far more interesting and complex than we thought,’ Zimin says.

The discovered counterexample raises fundamental questions for science about whether intuition can be relied upon, how critical thinking should be applied in mathematics, and how probabilistic evidence-based proofs should be interpreted.