Statement 1: In a group of 5 people always there will be 3 people who know each other or there will be 3 people who do not know each other.
The above statement is FALSE.

Consider the above relation graph among 5 people A, B, C, D and E. A and B know each other. So, they have a green line connecting them. A and C do not know each other. So, they have a red line connecting them and likewise for others. Now note that there is no green triangle in the above graph and there is no red triangle either. So, Statement 1 is false.

Statement 2: In a group of 6 people always there will be 3 people who know each other or there will be 3 people who do not know each other.
The above statement is TRUE.

Consider an arbitrary relation graph among 6 people A, B, C, D, E and F. I can see a red triangle between A, C and E and if you change any of the line between A, C and E you get a green triangle. There is a rigorous mathematical proof available in Wikipedia but I am skipping it.

In short we say R(3,3) = 6 and this is known as the Party Problem. In general, R(m,n) denotes the minimum value of k such that in any group k people always there will m people who know each other or there will be n people who do not know each other. These are called Ramsey Numbers after the British Mathematician Frank P Ramsey who proved that value of R(m,n) is finite for any choice of m and n.

It is very hard to find the exact values of R(m,n) for different values of m and n. We have plans to bring in quantum computers to find them. Below is the status till today.

I shall end with a famous story from Paul Erdos about computing Ramsey Numbers:
Erdős asks us to imagine an alien force, vastly more powerful than us, landing on Earth and demanding the value of R(5, 5) or they will destroy our planet. In that case, he claims, we should marshal all our computers and all our mathematicians and attempt to find the value. But suppose, instead, that they ask for R(6, 6). In that case, he believes, we should attempt to destroy the aliens.

 
QUESTION I: Why it is so difficult to compute the Ramsey Numbers? 
QUESTION II: How one would define R(i, j, k)?

REFERENCE: Ramsey's Theorem Wikipedia