The Ramsey number R(m,n) is the minimum size of a fully connected graph with edges colored red or blue such that there is either an m-sized blue clique or an n-sized red clique. R(3,3) is 6, which is stated as every group of 6 having 3 mutual friends or 3 mutual strangers. Ramsey showed that these exist for every m and n, I think.
But they grow very rapidly. Ramsey theory has usually the biggest constants found in mathematical papers.[1] There is a famous quote by Erdos:
"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."
But they grow very rapidly. Ramsey theory has usually the biggest constants found in mathematical papers.[1] There is a famous quote by Erdos:
"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."
[1]https://en.m.wikipedia.org/wiki/Graham%27s_number