Wow! R(k) < (4-c)^k for some c > 0. See gil kalai's blog. Was anyone at this talk, what value of c are they claiming, what methods were used?

# Cambridge Combinatorics Seminar

I can imagine how huge this would be if their proof is correct. Likely true, all four coauthors are extremely good mathematicians who have done fantastic works.

The field of extremal and probabilistic combinatorics has seen a lot of exciting developments recently hasn't it?

Wow! R(k) < (4-c)^k for some c > 0. See gil kalai's blog. Was anyone at this talk, what value of c are they claiming, what methods were used?

Looks like Tim Gowers was at the talk

he's teaching a piii course this semester, so he'd be in cambridge for the term which ends tomorrow.

Can someone explain to the unaware person (i.e. me) in the room why one should care about this result?

It's a famous open problem largely because the bound R(k) < 4^k (roughly) was proven close to 100 years ago by a very simple induction, taught in any first course in discrete math. This is the first time it's been improved to (4 - c)^k.

No-one thinks that's actually the right answer, which is probably 2^{k/2} or thereabouts, so probably one day long in the future the result under discussion will be forgotten. But this is certainly a very big breakthrough in this area, to add to several other big breakthroughs in extremal comb and related areas such as Kelley Meka on progressions, or Gilmer on union-closed.

Can someone explain to the unaware person (i.e. me) in the room why one should care about this result?

It's a famous open problem largely because the bound R(k) < 4^k (roughly) was proven close to 100 years ago by a very simple induction, taught in any first course in discrete math. This is the first time it's been improved to (4 - c)^k.

No-one thinks that's actually the right answer, which is probably 2^{k/2} or thereabouts, so probably one day long in the future the result under discussion will be forgotten. But this is certainly a very big breakthrough in this area, to add to several other big breakthroughs in extremal comb and related areas such as Kelley Meka on progressions, or Gilmer on union-closed.

This confirms my suspicion all along: The result is a nothing burger.

This confirms my suspicion all along: The result is a nothing burger.

Lol, tell me you don't know what you're talking about without telling me you don't know what you're talking about.

Huge result, may well end up in the annals. Looking forward to hearing about the proof.

Can someone explain to the unaware person (i.e. me) in the room why one should care about this result?

It's a famous open problem largely because the bound R(k) < 4^k (roughly) was proven close to 100 years ago by a very simple induction, taught in any first course in discrete math. This is the first time it's been improved to (4 - c)^k.

No-one thinks that's actually the right answer, which is probably 2^{k/2} or thereabouts, so probably one day long in the future the result under discussion will be forgotten. But this is certainly a very big breakthrough in this area, to add to several other big breakthroughs in extremal comb and related areas such as Kelley Meka on progressions, or Gilmer on union-closed.

Ah, so "old problem of Erdos" really means very, very old.

This is a really big result in combinatorics, not only Ramsey theory and not just another question of Erdos. Probably the biggest breakthrough since Szemeredi's theorem or something like this, if indeed it all turns out correct.

I don't think this is bigger than the 3AP breakthrough to be honest. The 3AP problem was also very old and one of top Erdos's problems. This Ramsey thing is huge but admit it the gap is still quite big.

Likely after this work there'll be a sequence of follow-up papers improving the upper bound by a little, just like what's happened with the union closed set problem, but unlikely the upper bound will even reach something like ${3}^{k}$.

This is a really big result in combinatorics, not only Ramsey theory and not just another question of Erdos. Probably the biggest breakthrough since Szemeredi's theorem or something like this, if indeed it all turns out correct.

No. It's an improvement from 4^k to 3.9999^k. Ok, so it took 100 years to get there, but it's not the biggest result in comb since Szemeredi, or even this year as pointed out by gjng above.

Noob question but: What’s the big open problem in Ramsey theory, and what implications does this new result have for it?

the ultimate goal is to obtain precise asymptotics of the diagonal ramsey number R(k) as k goes to infinity for a century the best that people could do is to prove that R(k) grows exponentially in k, i.e. some bounds like 2^(k/2)\leq R(k)\leq 4^k even an improvement at logarithmic scale is viewed as very good work, see e.g. the paper of Sah

This one is the first that truely improve the classical bound at the correct scale

This is a really big result in combinatorics, not only Ramsey theory and not just another question of Erdos. Probably the biggest breakthrough since Szemeredi's theorem or something like this, if indeed it all turns out correct.

No. It's an improvement from 4^k to 3.9999^k. Ok, so it took 100 years to get there, but it's not the biggest result in comb since Szemeredi, or even this year as pointed out by gjng above.

extreme combinatorics and additive combinatorics are two different subfields like algebraic numner theory and analytic number theory, so it's hard to compare the importance, for me both are big results

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