Question

Show that R(3, 4) ≥ 7 by showing that in a group of six people, where any two people are friends or enemies, there are not necessarily three mutual friends or four mutual enemies.

Short answer

There can’t be a set of four mutual enemies and there can’t be a set of three mutual friends in a group of six people \[R\left( {3,4} \right) \ge 7\].

Step-by-step solution

Given

In a group of six people, where any two people are friends or enemies there are not necessarily three mutual friends or four mutual enemies .

Explanation

Suppose that in a group of six people there are three men and three women.

Let people of same sex are always enemies and people of opposite sex are always friends.

Now any set of four people must include at least one men and one women.

Therefore a set of four mutual enemies cannot be possible and also set of three mutual friends can’t be possible.

Calculation

Therefore, there can’t be a set of four mutual enemies and there can’t be a set of three mutual friends in a group of six people.

Therefore,

\[R\left( {3,4} \right) \ge 7\]

Conclusion

\[R\left( {3,4} \right) \ge 7\]