The sixth Ramsey number is at most 147

The diagonal Ramsey number \(R(k)\) is the smallest order of a complete graph such that any \(2\)-coloring of its edges contain a monochromatic complete subgraph of order \(k\). It is well known that \(ak2^{(k/2)} < R(k) < 4^k / k^{(b*log(k))}\) for some absolute constants \(a>0\) and \(b>0\). On the other extreme, we know that \(R(3)=6\) and \(R(4)=18\), but already the exact value of \(R(5)\) is not known.

Determining the exact value of \(R(k)\) for \(k>4\) is a challenging problem, and a well known quote of Erdös says that if aliens invade the Earth and demand within a year the exact value of \(R(6)\), it would be better for the humans to fight the aliens. In this talk, we use the flag algebra method to show \(R(6)\) is at most \(147\) improving on the previous upper bound \(R(6) <= 165\).

This is a joint work with Sergey Norin and Jeremie Turcotte.