The ramsey-theory tag has no wiki summary.

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### Is every knot unavoidable in the embeddings of some graph?

Is it the case that, for any given knot $K$,
there exists some graph $G$ whose every embedding into $\mathbb{R}^3$
(or into $\mathbb{S}^3$)
contains a cycle that realizes $K$?
I know the ...

**5**

votes

**1**answer

339 views

### Are semigroups with finite-to-one right multiplication “moving”?

A semigroup $S$ is moving if $S$ is infinite, and for all finite
$F\subseteq S$ and infinite $A\subseteq S$, there are $a_{1},\dots,a_{k}\in A$ such that,
for all but finitely many $s\in S$,
$$
...

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votes

**1**answer

38 views

### Moving from positive upper Banach density to positive upper density [closed]

Under which conditions positive upper Banach density implies positive upper or lower asymptotic density? Thanks a lot.

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vote

**0**answers

89 views

### Mixed Tsirelson Norm

A couple of days ago I posted this question on Mathematics Stack Exchange. Surprisingly, so far, I haven't received any answers or comments about it (besides my own possible answer). Maybe I can get ...

**4**

votes

**1**answer

186 views

### Is Van der Waerden's function elementary

Van der Waerden's function was proved to have elementary upper bound on growth rate.
Is the Van der Waerden's function itself elementary in the sense of Kalmar?

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votes

**1**answer

592 views

### Big binary tree as an induced subgraph

I believe this is true:
Suppose $G$ is a graph. If $G$ has a subdivision of a large binary tree, prove that $G$ has an
induced subgraph which is a subdivision of a large binary tree or the line ...

**2**

votes

**1**answer

181 views

### Coloring of subgraphs of G^n

Let $G=(L,R,E)$ be a finite bipartite graph, such that for each $v\in L\cup R: deg(v)>0$. Define $E^{(n)}=\{(\overline{l},\overline{r}) | \overline{l}=(l_1,...,l_n)\in L^n , ...

**2**

votes

**1**answer

261 views

### Geometric van der waerden theorem

Van der Waerden theorem states that sufficiently long initial segment of the natural numbers when divided into $r$ parts contains an arithmetic progression of length $k$. The length of the initial ...

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votes

**1**answer

104 views

### Multipartite Ramsey theorem

Given $c<\infty$ colors, positive integers $k_1,\dots,k_n$ and positive integers $N_1,\dots,N_n$. Then there exist positive integers $M_1,\dots,M_n$ so that for disjoint finite sets $A_1,\dots,A_n$ ...

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votes

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120 views

### Is There a Graph that is Ramsey for $P_{2n}$ but is $C_{2n+1}-$free

Write $F\to G$ to mean that for every two coloring of the edges of $F$, there exists a monochromatic copy of $G$. Nesetril and Rodl proved that for a graph $G$, there exists a graph $F\to G$ with ...

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415 views

### Deriving Konig's Lemma directly from Infinite Ramsey's Theorem for triples

Let KL denote König's Lemma (for trees over $\mathbb{N}$), and RT(3) denote the
Infinite Ramsey Theorem for triples over $\mathbb{N}$ (notation as in Simpson's
book Subsystems of second order ...

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vote

**1**answer

139 views

### Ramsey numbers and graph spectra

I am interested in graph Ramsey theory. Are there any papers which investigate Ramsey numbers $R(G,G)$ of an arbitrary graph by analyzing the spectrum of $G$? In general, has anyone found any ...

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vote

**1**answer

95 views

### Distribution of Induced Subgraphs of Extremal Ramsey Graphs

Choose $k$. Let $G = (V,E)$ be a graph on $n = R(k,k)-1$ vertices (that is, $G$ is an extremal example for $R(k,k)$, and $g : E \to \{r, b\}$ be an edge 2-coloring such that there is no monochromatic ...

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465 views

### Combining van der Waerden's theorem with Ramsey's theorem

Consider positive integers $c$, $k$, and $s$. Does there exist some $N = N(c,k,s)$ such that the following holds?
Take any $c$-coloring of the $k$-tuples of integers in $[1,N]$. Then there is an ...

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votes

**1**answer

178 views

### What is a Ramsey Graph? [closed]

Can anyone explain me what is a Ramsey Graph with a simple example?
What are its properties?

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vote

**1**answer

213 views

### Van der Waerden's Theorem Over $\mathbb{Z}_p$

Define a length $k$ arithmetic progression in $\mathbb{Z}_p$ to be a set of the form
$\{ax+b : x \in [k]\}$ with $a \in \mathbb{Z}_p^*$ and $b \in \mathbb{Z}_p$.
Let $HJ(k, c)$ be the Hales-Jewett ...

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votes

**3**answers

825 views

### A General Framework for Ramsey Theory ?

There are few results in modern mathematics that I find so deep and full of philosophical implications as Ramsey's theorem.
I am aware (at some basic level) that it has generated a plethora of ...

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**1**answer

2k views

### Reconstructing the argument that yields Graham's number

Graham's number achieved a kind of cult status, thanks to Martin Gardner, as the largest finite number appearing in a mathematical proof. (It may no longer hold that record, but that is not my concern ...

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votes

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351 views

### From very many sets of fixed measure in a probability space, can we select many that have a positive intersection?

I assume the following Lemma is either well known or, more probably, a
Corollary of a much stronger well known Theorem, and I would be grateful for a
reference:
For all $\delta\in (0,1)$ and all ...

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votes

**1**answer

704 views

### Graham-Rothschild via Hales-Jewett

I am currently reading the recent preprint of Dodos, Kanellopoulos, Tyros, where the ambitiously short proof of Density Hales Jewett theorem is provided. The important ingredient is Graham-Rothschild ...

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**1**answer

389 views

### small Ramsey number and Brooks' Theorem

I'm studying about Graph Ramsey Theory now. Starting this study, I'm reading Chvatal and Harary's series of papers. In the second paper (V.Chvatal, F.Harary, Generalized ramsey theory for graphs,Ⅲ. ...

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votes

**0**answers

123 views

### Sparse ramsey theory

It is known that for any graph H and all $k∈N$, there exists a graph $G$ such that any $k$-coloring of the edges of $G$ yields a monochromatic copy of H and ω(G)=ω(H) (the two graphs have the same ...

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**1**answer

173 views

### A Ramsey-like lower bound?

Does there exist a graph $G$ which cannot be properly vertex-coloured with 3 colours (i.e. $G$ has chromatic number at least 4), such that for every graph $H$, if $H$ contains a triangle but there ...

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### Weak Arithmetic Progressions

I am studying a special type of a sequence on the naturals which I am calling a weak arithmetic progression.
Formally I call a k-sequence $x_1< x_2 \cdots< x_k$ a weak arithmetic progression ...

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175 views

### Van der Waerden like theorem

I am trying to develop bounds for the function B(k) where B(k) is defined as the least such positive integer so that whenever the set $\{1,2,\cdots B(k)\}$ is partitioned into two parts at least one ...

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687 views

### Hales Jewett Theorem

In the book Ramsey Theory by Graham, Rothschild and Spencer the authors state:
The Hales Jewett Theorem strips van der Waerden's theorem of its unessential elements and reveals the heart of Ramsey ...

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votes

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452 views

### The Problem about 2-coloring finite plane

Suppose we color a $X \times X$ finite plane by red and blue arbitrarily. How large does X need to be to guarantee a monochromatic combinatorial square $k \times k$
1 0 1 0 1 1
1 1 1 1 1 1
1 ...

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vote

**1**answer

378 views

### Combinatorial optimization and graph coloring

I am considering the following problem:
(i) Fix $n$ and color the edges of $K_n$ red and blue arbitrarily.
(ii) Let $M$ be the set of monochromatic triangles in $K_n$ and define $g:M\rightarrow ...

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232 views

### Slightly improving bounds on two-color Ramsey numbers by globally pruning edges and counting connected vertices in instances of two-colored complete graphs

The two-color Ramsey number, $R(m, n)$, is the minimum number of vertices, $||V||$, in a complete graph necessary for there to exist a clique of order $m$ or an independent set of order $n$. In terms ...

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308 views

### Extracting countable chains from linear orders

There is a well-known fact in infinite combinatorics asserting that for each infinite linear order $P$ there is a countable subset $R\subseteq P$ of order type either $\omega$ or $\omega^{*}$
(by ...

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vote

**2**answers

372 views

### Quasi-Ramsey Graphs

A graph with number of vertices less than R(n,n)-1, is called quasi-Ramsey of case n, if it has no complete graph Kn in itself or its complementary graph, and if added another vertex, no matter how it ...

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735 views

### Density Ramsey theorems with explicit asymptotics

I wonder what interesting and non-trivial examples of density Ramsey theorems with explicit asymptotics are there?
I'm aware of two examples: Szemerédi's theorem and density Hales-Jewett theorem.
...

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716 views

### Is this Ramsey-type problem an open problem?

A blog claims that the following Ramsey-type (or van der Waerden type) problem is open:
If the natural numbers are colored with finitely many colors, must there exist x and y (not both 2) such that ...

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198 views

### Constructive lower bounds for multicolor Ramsey numbers

The $k$-color Ramsey number of the complete graph $K_n$, denoted with $R_k(n)$, is defined to be the smallest integer $t$, such that in any $k$-coloring of the edges of $K_t$, there is a complete ...

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273 views

### 3x3 submatrix with only $0$ or $1$ entries

I decided to cross-post the question here from math.stackexchange.com because I got no answer from there.
It is a quick question on bipartite Ramsey numbers (I'm not an expert on the subject, so ...

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103 views

### Bound on how many 2-colored graphs have maxredclique $i$, maxblueclique $j$

Fix $i,j$. I want a bound $F(i,j)$ on the following:
$\sum_{n=1}^\infty$ (the number of 2-colorings of the edges of $K_n$ such that
the MAX RED clique is size exactly ...

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votes

**2**answers

581 views

### Partition calculus question

For $m,n,k < \omega$, consider the equation
$X \to (\omega \times k)^{m}_{n}$
What is the smallest $X$ known to satisfy it?
Baumgartner-Hajnal theorem gives a satisfactory answer for $m=2$, but ...

**3**

votes

**1**answer

386 views

### Permutations of Grid Colorings

Hello all,
This question arises out of a Ramsey-theoretic study related (but tangential) to this question on CSTheory: Grid $k$-coloring without monochromatic rectangles.
Background:
In general, we ...

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votes

**2**answers

617 views

### Asymptotics for Ramsey Theory

Ramsey Theory says that every sufficently large (but finite) complete graph having $d-$coloured edges contains a monochromatic complete subgraph with $k$ vertices.
One could ask for asymptotics: Let ...

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votes

**3**answers

610 views

### Arithmetic progressions of length 3 in subset of Z_n of size n^d

Let $A\subset\mathbb{Z}/n\mathbb{Z}$ such that: $|A|>n^{d}$ ($0< d <1$).
Let $C=\{(x,y,2y-x)\in A\times A \times A\}$ be the set of $3$-term arithmetic progressions within $A$.
[The ...

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2k views

### Cliques, Paley graphs and quadratic residues

A question I've thought about, on and off for a long time, is how to improve the best bounds that (seem to be) known for the clique numbers of Paley graphs.
If p=1 mod 4 is a prime, we can define the ...

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votes

**1**answer

555 views

### Best lower bound for off-diagonal Ramsey numbers

What are the current best lower bounds for off-diagonal Ramsey numbers $R(k,l)$ with $l$ of order unity and asking for asymptotic behavior for large $k$, such as $R(k,4)$, $R(k,5)$, and so on? ...

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467 views

### Ergodic limits along subsets of $\mathbb{N}.$

Let say that an infinite subsets $A$ of $\mathbb{N}$ is "nice w.r.to ergodic limits", if it can replace $\mathbb{N}$ in the individual ergodic theorem, that is, if it is such that the following ...

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### Differences of near diagonal Ramsey numbers.

I am a graduate student trying to get involved in Ramsey theory. My question comes from:
Erdős on graphs: his legacy of unsolved problems
By Fan R. K. Chung, Paul Erdős, Ronald L. Graham
p.14 of ...

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**0**answers

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### Any 2-coloring of the scope of equilateral triangle must create a monochromatic rectangular triangle ?

Hello All,
I'm trying to prove something that we know that is correct but can't find how to do it or any example to show how to solve it.
Any 2-coloring of the scope of equilateral triangle must ...

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**1**answer

460 views

### monochromatic cycle-free colouring of the complete graph on R?

Hi
So there is an edge-colouring of a complete graph on R (the reals), with countably many colours that as no monochromatic triangle. To find it map R to (0,1) write the numbers in binary and if 2 ...

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1k views

### Monochromatic triangles in every two-coloring of the plane?

An old problem (possibly due to Erdős and Graham?): given a triangle $T$ and a two-coloring of the plane, does there necessary exist a monochromatic congruent copy of $T$? Here "monochromatic" means ...

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votes

**1**answer

314 views

### Ramsey pairs of classes graphs

Motivation
Call a set $H$ of vertices of a graph $G$ homogeneous if it is either independent or complete.
Every perfect graph of size $n$ has a homogeneous subset of size $\sqrt n$.
Noga Alon has ...

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votes

**2**answers

1k views

### Proofs of Lower Bounds for Ramsey Numbers?

As a sort of dual question to this question, I am wondering what proofs people know of lower bounds on Ramsey numbers $R(k, k)$. I know of two proofs: there is Erdos's beautiful probabilistic ...

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1k views

### Noncombinatorial proofs of Ramsey's Theorem?

I know of 2(.5) proofs of Ramsey's theorem, which states (in its simplest form) that for all $k, l\in \mathbb{N}$ there exists an integer $R(k, l)$ with the following property: for any $n>R(k, l)$, ...