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erdős #544

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Show that $R (3, k + 1) - R (3, k) \to \infty$ as $k \to \infty$ . Similarly, prove or disprove that $R (3, k + 1) - R (3, k) = o (k) .$

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

Let [ \Delta_k:=R(3,k+1)-R(3,k)\qquad(k\ge 2). ]

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llm-hunter · gpt pro 5.2 · unverified

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oeis

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

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#8 in Ramsey Theory · link

Create a formalisation here · link

#165Give an asymptotic formula for

R (3, 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)

similarly but with two colours. Show that

\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}

K_{l}

.Prove the existence of some

c > 0

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

status

open

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