Seonglae ChoThis page summarizes recent progress on improving upper bounds for the diagonal Ramsey numbers .
The authors achieved a historic exponential improvement on the long-standing general upper bound for . Since Paul Erdős’s 1935 bound had remained essentially unchanged for nearly a century, this work shows that for all sufficiently large
breaking the “barrier” at 4 for the first time.
At a high level, the proof develops a new density-increment–style argument and leverages a structured “red book” framework to obtain the improved constant in the exponent.
An exponential improvement for diagonal Ramsey
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...
https://arxiv.org/abs/2303.09521
www.quantamagazine.org
https://www.quantamagazine.org/math-leap-exponentially-improves-ramsey-number-bound-20230502/