Question

Show that if \(n\) is an integer with \(n \ge 2\), then the Ramsey number \(R(2,n)\) equals \(n\).

Short answer

The resultant answer is \(R(2,n) = n\).

Step-by-step solution

Given data

The given expression \(n\) is an integer with \(n \ge 2\).

Concept of Ramsey number

Ramsey number\(R(m,n)\)denotes the minimum number of people at a party such that there are either\(m\)mutual friends or\(n\)mutual enemies (\(m \ge 2,n \ge 2\)).

Assume \(n\)people at the party

Let there be \(n\) people at the party.

If there are 2 mutual friends, then there are 2 mutual friends or \(n\) mutual enemies (since the first statement is true).

If there are not 2 mutual friends, then the \(n\) people all have to be mutual enemies and thus there are 2 mutual friends or \(n\) mutual enemies (since the second statement is true).

It notes then that there are either 2 mutual friends or \(n\) mutual enemies at the party and thus; \(R(2,n) \le n\).

Simplify first and second equation

Let \(k\) be a positive integer with \(k < n\) and let there be \(k\) people at the part.

If none of the people are mutual friends, then it has \(k\) mutual enemies and thus the statement "2 mutual friends or \(n\) mutual enemies" cannot be true.

It has then obtained a contradiction and thus such a \(k\)-value cannot exist, which means that \(n\) has to be the minimum value, \(R(2,n) = n\).

Hence, the given expression \(n\) is an integer with \(n \ge 2\).