I understand why R(3,3) is equal to 6 since we have a clique of 3 in a graph of 6 vertices. What I don't understand is that by definition of Ramsay number it says that for a R(r,s) we have to find a clique of either 3 or 4 in size. In this case, we can find a clique of 3 in a graph of 6 nodes unconditionally. Could someone explain to this hopeless student why R(3,4) = 9?
2026-03-28 15:45:58.1774712758
why is the Ramsey number $R(3,4)$ not equal to $R(3,3)$?
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For $R(3,4)$ you must find either a clique of size $3$ or an independent set of size $4$. The graph
on $6$ vertices has neither a clique of size $3$ nor an independent set of size $4$. Every set of $3$ vertices contains two that are not adjacent, and every set of $4$ vertices contains two that are adjacent.