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The question of what is an Erdős problem and what is not is not black and white. My team has managed to solve a problem that's sort of an Erdős problem. In the
by skinner_ 2y ago
The question of what is an Erdős problem and what is not is not black and white. My team has managed to solve a problem that's sort of an Erdős problem. In the early sixties, Leo Moser asked about the value of some quantity. In a 1985 paper, Erdős commented that the quantity “seems likely” to be less than 1/4. It was known to be at most 2/7. After a long line of improvements by several teams, we were the first ones to go below 1/4.
For those who are interested, https://www.sfu.ca/~vjungic/RamseyProjects/section-11.html https://www.sfu.ca/~vjungic/RamseyProjects/section-11.html describes the question. Our answer is at https://arxiv.org/abs/2207.14179 https://arxiv.org/abs/2207.14179, and a popsci account of our result is at https://www.quantamagazine.org/mathematicians-break-bounds-in-coloring-problem-20230719/ https://www.quantamagazine.org/mathematicians-break-bounds-i... .
- throw_pm23 2y agoNice... but was 1/4 here a kind of arbitrary "nice number" threshold? It seems now the final value is somewhere in [0.22, 0.247], right?
- skinner_ 2y agoVery good question! Yes, the final value is somewhere in there. I expect that it's the lowest point, meaning Croft's Tortoise is optimal. Since then we have an unpublished new 0.241 upper bound with the same proof but wider computer search. We don't know what Erdős had in mind when picking 1/4, but we know something that seems to make the 1/4 value special. Many of the earlier attempts to prove Erdős's conjecture were based on a notion called fractional chromatic number. The numbers went down like this: 0.2857, 0.2813, 0.2763, 0.2565, 0.2518, 0.2506. We now have a new preprint that reaches exactly 1/4, not more not less, and we don't know how to improve it: https://arxiv.org/abs/2311.10069 https://arxiv.org/abs/2311.10069 So it seems like the fractional chromatic number based approach has an inherent barrier at 1/4. If that is true, those earlier fractional chromatic number based attempts were doomed, and we only managed to break that barrier because we used some extra ideas. (Namely, Fourier analysis).
- throw_pm23 2y agoThanks for explaining, and again, nice set of results!
- TFBloom 2y agoAdded! https://www.erdosproblems.com/232 https://www.erdosproblems.com/232 There are certainly many Erdős problems, broadly interpreted, still to be added. Any suggestions of missing problems to be added are welcome.
- skinner_ 2y agoAwesome, thank you! A seal of approval!