The bunkbed conjecture is false
The bunkbed conjecture has been disproven after decades, with a counterexample using a 3-hypergraph. The authors emphasize transparency in research and question the acceptance of probabilistic results in publishing.
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The bunkbed conjecture (BBC), which posited that the probability of connecting two vertices on the same level of a product graph is greater than or equal to that of connecting vertices on different levels, has been proven false. Initially conjectured by Kasteleyn in the early 1980s, the BBC remained unproven for decades, despite its seemingly obvious nature. Igor Pak, along with graduate students Nikita Gladkov and Alexandr Zimin, developed a counterexample using a specific 3-hypergraph structure. Their findings revealed that the probability difference between paths connecting vertices on the same level versus different levels is extremely small but negative, thus disproving the conjecture. The authors faced challenges in their computational experiments, which initially failed to yield a clear counterexample. They emphasized the importance of rigorous testing and the need for transparency in both successes and failures in mathematical research. The paper discusses broader implications regarding the acceptance of probabilistic results in mathematical journals, questioning whether journals should accept findings based on high confidence levels rather than absolute certainty. Ultimately, the authors advocate for a culture of openness in sharing both successful and unsuccessful research efforts.
- The bunkbed conjecture has been proven false after decades of belief in its validity.
- A counterexample was developed using a specific 3-hypergraph structure.
- Initial computational experiments failed, highlighting the challenges in disproving established conjectures.
- The paper raises questions about the acceptance of probabilistic results in mathematical publishing.
- The authors advocate for transparency in sharing both successful and unsuccessful research outcomes.
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- Some users question the mathematical rigor of the counterexample, suggesting issues with the definitions used.
- There is a discussion about the intuition behind the conjecture, particularly how it may hold true for smaller cases but fail for larger ones.
- Several commenters emphasize the importance of probabilistic results in guiding research, even if they do not constitute formal proofs.
- Users express a desire for clearer explanations and visual aids to better understand the concepts involved.
- Overall, there is a mix of surprise and intrigue regarding the counterintuitive nature of the findings.
12 commentsSometimes we tend to have this intuition that if a rule applies to all low numbers, than it must apply to all numbers, that there can't be some huge number after which it breaks down (unless of course it explicitly includes that huge number, such as the rule "all numbers are smaller than a billion billion billion").
This is such a powerful intuition, even though it's obviously wrong, that rules that break it are even seen as a problem. For example there is the so-called "hierarchy problem" in physics, which states something like "there is no reason why the weak force is so much stronger than gravity". As if 2 times as strong is somehow fundamentally different than it being 10^24 times as strong from a mathematical perspective. And this has ended up being called a major problem with the standard model, even though it's completely normal from a mathematical perspective.
Reading that and then rereading the post, it all made a whole more sense: why the conjecture is intuitively appealing and why the computational approach doesn't readily result in a proof.
Ed: reflecting on my first sentence, I guess I tend to think of proofs as being at least as important a product of math as the raw truth of a statement. A fact is a fact, but a proof represents (some level of) understanding, and that's the good stuff. Experiments are potentially good science, but usually not math.
Could someone explain why the conjecture seemed obviously true? I have no background with it, but just reading the description here, I was caught off guard by the sentence. What made it seem obvious?
The counterexample seems to rely on correlations between edge deletions which makes no sense because the deletions should be independent (in the definition I'm seeing on Wikipedia).
I could be wrong here because I haven't read it in detail but on first sight, it looks like there are some serious issues with mathematical rigour here.
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