Let $R (3, 3, n)$ denote the smallest integer $m$ such that if we $3$ -colour the edges of $K_{m}$ then there is either a monochromatic triangle in one of the first two colours or a monochromatic $K_{n}$ in the third colour. Define $R (3, n)$ similarly but with two colours. Show that $\frac{R ( 3 , 3 , n )}{R ( 3 , n )} \to \infty$ as $n \to \infty$ .

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R (3, k)

R (3, k + 1) - R (3, k) \to \infty

k \to \infty

R (3, k + 1) - R (3, k) = o (k) .

s \geq 3

R (s, k) ≫ \frac{k ^{s - 1}}{( lo g k ) ^{c}}

c = c (s) > 0

R (k, l)

n

K_{n}

K_{k}

K_{l}

c > 0

k \to \infty lim \frac{R ( k + 1 , k )}{R ( k , k )} > 1 + c .

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