erdős #166

← #165 · #167 → (packet.json; erdosproblems.com)

Prove that $R (4, k) ≫ \frac{k ^{3}}{( lo g k ) ^{O (1)}} .$

unreviewedOpen. Worked here; no verified result yet.

graph theory · solved · prize $250 · formalized (Lean) · 0 attempts

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Formal proof

AMS 5 · open (literature)

theorem erdos_920 :
    answer(sorry) ↔ ∀ k : ℕ, k ≥ 4 → ∃ c > 0,
      (fun n ↦ f k n) ≫ (fun n ↦ (n : ℝ) ^ (1 - 1 / ((k : ℝ) - 1)) / (log n) ^ c)

formal-conjectures/166.lean ↗

OEIS1

A059442 — Array of Ramsey numbers R(n,k) (n >= 2, k >= 2) read by antidiagonals.2,3,3,4,6,4,5,9,9,5,6,14,18,14,6,7,18,25,25,18,7,8,23

Connections2

#5 in Ramsey Theory · link

Create a formalisation here · link

#77If

R (k)

is the Ramsey number for

K_{k}

, the minimal

n

such that every

2

-colouring of the edges of

K_{n}

contains a monochromatic copy of

K_{k}

, then find the value of

k \to \infty lim R (k)^{1/ k} .

A059442 #78Let

R (k)

be the Ramsey number for

K_{k}

, the minimal

n

such that every

2

-colouring of the edges of

K_{n}

contains a monochromatic copy of

K_{k}

.Give a constructive proof that

R (k) > C^{k}

for some constant

C > 1

.A059442 #87Let

ϵ > 0

. Is it true that, if

k

is sufficiently large, then

R (G) > (1 - ϵ)^{k} R (k)

for every graph

G

with chromatic number

χ (G) = k

? Even stronger, is there some

c > 0

such that, for all large

k

R (G) > c R (k)

for every graph

G

with chromatic number

χ (G) = k

?A059442 #545Let

G

be a graph with

m

edges and no isolated vertices. Is the Ramsey number

R (G)

maximised when

G

is 'as complete as possible'? That is, if

m = (2 n) + t

edges with

0 \leq t < n

then is

R (G) \leq R (H),

where

H

is the graph formed by connecting a new vertex to

t

of the vertices of

K_{n}

?A059442 #812Is it true that

\frac{R ( n + 1 )}{R ( n )} \geq 1 + c

for some constant

c > 0

, for all large

n

? Is it true that

R (n + 1) - R (n) ≫ n^{2} ?

A059442 #986For any fixed

s \geq 3

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

for some constant

c = c (s) > 0

.A059442 #1029If

R (k)

is the Ramsey number for

K_{k}

, the minimal

n

such that every

2

-colouring of the edges of

K_{n}

contains a monochromatic copy of

K_{k}

, then

\frac{R ( k )}{k 2 ^{k /2}} \to \infty.

A059442 #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 .

A059442

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