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.
Very 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 Since then we have an unpublished new 0.241 upper bound with the same proof but wider computer search.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
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).
> Ron Graham bet him $500 that he could not stop taking the drug for a month. Erdős won the bet, but complained that during his abstinence mathematics had been set back by a month: 'Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper.' After he won the bet, he promptly resumed his amphetamine use.
I clicked on a random open problem and found https://www.erdosproblems.com/170. The text of the problem didn't make sense. The wikipedia article stated the problem much more nicely which was thankfully linked from the OP's website.
Wish the problem could be stated more simply on the main site as well. Will certainly try and help the owner of the website.
For those who are interested, https://www.sfu.ca/~vjungic/RamseyProjects/section-11.html describes the question. Our answer is at https://arxiv.org/abs/2207.14179, and a popsci account of our result is at https://www.quantamagazine.org/mathematicians-break-bounds-i... .
There are certainly many Erdős problems, broadly interpreted, still to be added. Any suggestions of missing problems to be added are welcome.
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
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).
http://en.wikipedia.org/wiki/Paul_Erd%C5%91s
Wish the problem could be stated more simply on the main site as well. Will certainly try and help the owner of the website.
Of course, what the clearest explanation is is very much dependent on one's own background and personal tastes!
For example for this problem I think a lot about sumsets and difference sets anyway, so it made sense to phrase it in those terms.
If you have a suggestion on how it could be rephrased for clarity then please do email in.
https://en.wikipedia.org/wiki/Proofs_from_THE_BOOK