Problem $1(20/100)$

Recall that $R(n,n)$ is the smallest number $N$ such that regardless how one colors the edges

of $K_N,$ there is always a monochromatic $K_n$ in the colored $K_N.$ We also saw that $R(3,3)=6.$

Suppose aliens invade the earth and threaten to obliterate it in a year's time unless human

beings can find the Ramsey number for red five and blue five. We could marshal the

world's best minds and fastest computers, and within a year we could probably calculate

the value. If the aliens demanded the Ramsey number for red six and blue six, however,

we would have no choice but to launch a preemptive attack. $--$ Paul Erdos

To check whether $R(6,6)=m,$ one has to go over all possible edge coloring of $K_m$

How many different edge colorings are there on $K_m?$ What is the number for $m=110?$

Suppose your computer is so smart that it detects monochromatic triangle in $1$ CPU cycle

$($apparently surreal, in real life not even a quadratic time algorithm is known, see e$.$g$.$

here$).$ How much time would it take to go over all the colorings? CPU frequency is $3\mathrm{.}2$ GHz