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 $ak{2}^{(k/2)}<R(k)<4^k/k^{(b\ast log(k))}$ 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.