R(3,3) doesn't say a clique necessarily forms. It says a clique forms or an anti-clique forms (or both). If all six are friends, then there are (6 choose 3) cli

R(3,3) doesn't say a clique necessarily forms. It says a clique forms or an anti-clique forms (or both). If all six are friends, then there are (6 choose 3) cliques. If all six hate each other, there are (6 choose 3) anti-cliques. If one kid is hated by everyone, that in some sense decreases the odds of a clique but simultaneously increases the chances of an anti-clique, and R(3,3)=6 guarantees that in some sense these increases/decreases balance each other out.