erdős #165

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Give an asymptotic formula for $R (3, k)$ .

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graph theory · open · prize $250 · 0 attempts

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gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let (R(3,k)) be the least (n) such that every red/blue colouring of (E(K_n)) contains either a red triangle or a blue (K_k) (equivalently: every triangle‑free graph on (n) vertices has an independent set of size (k)). ([arXiv][1])

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oeis

A000791 — Ramsey numbers R(3,n).1,3,6,9,14,18,23,28,36

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#544Show that

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

k \to \infty

. Similarly, prove or disprove that

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

A000791 #553Let

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)

\frac{R ( 3 , 3 , n )}{R ( 3 , n )} \to \infty

n \to \infty

.A000791 #986For any fixed

s \geq 3

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

for some constant

c = c (s) > 0

.A000791 #1030Let

R (k, l)

be the usual Ramsey number: the smallest

n

such that if the edges of

K_{n}

are coloured red and blue then there exists either a red

K_{k}

or a blue

K_{l}

.Prove the existence of some

c > 0

such that

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

A000791

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