The ramsey-theory tag has no wiki summary.

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### Could there be an exact formula for the Ramsey numbers?

Let $R(k)$ denote the diagonal Ramsey number, i.e. the minimal $n$ such that every red-blue colouring of the edges of $K_n$ produces at least one monochromatic $K_k$.
Is it possible that there ...

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### Does anyone want to see the critical figure for n= 7? [closed]

I watched Ronald Lewis Graham's youtube blurb for the "happy ending problem". it's about 5 minutes long. I was able to supply him with the positions of the points for the case: n=7. it looks like a ...

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### 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 $n>R(k, l)$, ...

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### Why have most maximal cliques of Paley graphs odd size?

I ask this question mainly by curiosity.
See here for definitions and a plot of the clique numbers of the Paley graphs for the primes $p\equiv 1 \pmod 4$ up to $10000$.
Is there an ...

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**3**answers

793 views

### A stronger version of Van der Waerden's theorem?

Let $W$ be an infinite word over a finite alphabet $\{1,\dots,n\}$ and $k$ a positive integer. An easy application of Van der Waerden's theorem implies the existence of $k$ disjoint and consecutive ...

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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 ...

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**4**answers

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### 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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476 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 ...

**5**

votes

**1**answer

367 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

599 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 ...

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128 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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**0**answers

47 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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**0**answers

94 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

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### 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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### Magnitude of Graham's Number?

I recently stumbled across this number, and then (foolishly, most likely) decided to try to describe it in a blog post
http://frothygirlz.com/2010/01/14/big-numbers-part-2/
Q - Are there any ...

**2**

votes

**1**answer

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### 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 , ...

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### Applications of infinite Ramsey's Theorem (on N)?

Finite Ramsey's theorem is a very important combinatorial tool that is often used in mathematics. The infinite version of Ramsey's theorem (Ramsey's theorem for colorings of tuples of natural numbers) ...

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**3**answers

476 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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votes

**2**answers

660 views

### Infinite Ramsey theorem with infinitely many colours

Clearly, it is possible to colour the edges of an infinite complete graph so that it does not contain any infinite monochromatic complete subgraph. Now what about the following?
Let $G$ be the ...

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**1**answer

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### 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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**1**answer

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### 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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**1**answer

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### 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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856 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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### 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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**2**answers

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### 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

141 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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**1**answer

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### 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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**1**answer

223 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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### 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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### 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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### 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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### 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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### 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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### 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

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### 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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### 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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### 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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### 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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### 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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**2**answers

487 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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### Erdos Conjecture on arithmetic progressions

Introduction:
Let A be a subset of the naturals such that $\sum_{n\in A}\frac{1}{n}=\infty$. The Erdos Conjecture states that A must have arithmetic progressions of arbitrary length.
Question:
I ...

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**1**answer

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### 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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### 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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### 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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### 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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**1**answer

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### 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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### 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

587 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 ...

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votes

**3**answers

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### 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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### 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 ...