4 ms·

> Can we improve 3.993 to 3.9? Maybe to 3.4? And what about 3?” Pi is feeling a little left out. If that turns out to be the true asymptotic behavior of Ramsey

• OJFord 3y ago
  I thought it was entertainingly specific, and fortunately the abstract at least [0] was slightly less immediately beyond me than I feared, to satisfy my curiosity a bit: The main proof is an improvement to the 4^k bound (standing since 1935) to (4-eps)^k for some epsilon. They additionally prove (I guess they have properties that make it slightly easier than other rounder/bigger numbers?) it for eps=2^-10 and eps=2^-7 specifically. (3.993 then comes from the latter, 4 minus it gives 3.9921 and change, but of course you need to round that up to 3.993 in order to say it's a bound: it's not definitely less than 3.992, since it could lie between the two.) So yes maybe/it probably can be improved from 3.993, because that's a bit of a tangential claim anyway - the main thing is that it's 'some non-zero amount less than 4'^k. (But mostly yes it was beyond me, I won't pretend to be able to even attempt to understand the proof really.) [0]: https://arxiv.org/abs/2303.09521 https://arxiv.org/abs/2303.09521