## Some News from a Seminar in Cambridge

### On an old problems of Erdős

(h/t Michael Simkin and Nati Linial)

Here is a somewhat mysterious announcement for a combinatorics seminar lecture at Cambridge.

Which old problems of Erdős are we talking about? Here is a picture from the seminar itself.

Stay tuned (or try to test your intuition or guess in the comment section.)

Nothing yet here.
The paper is now on the arXiv.

## An exponential improvement for diagonal Ramsey

Marcelo Campos, Simon Griffiths, Robert Morris, and Julian Sahasrabudhe

The Ramsey number $R(k)$ is the minimum $n \in \mathbb N$ such that every red-blue colouring of the edges of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $K_k$. We prove that

$R(k)\le (4-\epsilon)^k$

for some constant $\epsilon$. This is the first exponential improvement over the upper bound of Erdős and Szekeres, proved in 1935.

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### 9 Responses to Some News from a Seminar in Cambridge

1. rosskang says:

Wow.
It’s a fitting place to make such an announcement.

2. Lior Pachter says:

👀

3. ryeguy10 says:

Amazing! I am very excited to join the Cambridge combinatorics group next year.

4. Mike says:

Is that Tim Gowers in the photo?

• He was in attendance at any rate (I glean from Twitter).

5. Mike says:

I just started to skim through the paper. I’m reminded, in exploiting the existence of large books in estimating various Ramsey numbers, that Thomason had some thoughts about this approach. The article was in one of the (greenbook) LMS monographs for the BCCs back in the late 80s or early 90s and I can’t recall the details now (due to failing memory). I’d be interested in his thoughts here.
Truly this is a virtuoso paper, well worthy of study.

6. This is brilliant! There is also a new development on explicit constructions: https://arxiv.org/abs/2303.06802

7. Gil Kalai says: