**4**

votes

**3**answers

1k 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 ...

**3**

votes

**1**answer

107 views

### An extremal combinatorics problem involving column summation

Given $n\in\Bbb N$, $\alpha>0$, $\beta\in\big[\frac12,1\big]$ denote $\mathcal R_{n,\alpha,\beta}$ as collection of all $2^n\times 2^{n^\alpha}$ $0/1$ matrices with every row summing to strictly ...

**1**

vote

**2**answers

133 views

### question about literature in the field of Ramsey's theory [closed]

i am searching for an on- line paper or a book, or maybe just a paper or a book which consists a proof of finite Ramsey's theorem for sets (not for graphs). i need a combinatorial proof which is not ...

**6**

votes

**0**answers

192 views

### The Hales-Jewett Theorem for an infinite alphabet

Recall the Hales-Jewett Theorem:
HJT: Given a finite alphabet $A$ and some $r \in \mathbb{N}$, there is some $H \in \mathbb{N}$ such that whenever $A^H$, the set of all length-$H$ words from $A$, ...

**6**

votes

**0**answers

96 views

### Is there a Ramsey theory for Kneser graphs?

Ramsey theory for graphs usually studies colorings of the edges of complete graphs. I'm interested whether there are any results about edge-colorings of Kneser graphs. More specifically, I'm most ...

**4**

votes

**1**answer

72 views

### Show existence maximal clique of order $s$ in an multigraph where each vertex is colored with a set of colors

You are given a multigraph $G$ with $n$ vertices as follows:
$V := (v_1, v_2, \dots ,v_n)$
$C := \{c_1, c_2, \dots\}$, be an infinite set of colors.
$f: V \rightarrow \mathbb{P}_{\le m}(C) $, a ...

**3**

votes

**2**answers

110 views

### Partition regular systems: do they have solution in (very dense) set of integers?

A partition regular system is a linear system of equations of the form $A\cdot x=0$, which satisfies a Ramsey-type result (namely, that for each $r>0$ whenever we colour the integers in $r$ ...

**14**

votes

**1**answer

315 views

### Convergence rate of Fagin's 0-1 law for first-order properties of random graphs

Fagin's 0-1 law for first-order properties of random graphs states that, for every first-order sentence in the logic of graphs, the probability that a uniformly random $n$-vertex graph models the ...

**11**

votes

**5**answers

1k views

### Can one make Erdős's Ramsey lower bound explicit?

Erdős's 1947 probabilistic trick provided a lower exponential bound for the Ramsey number $R(k)$. Is it possible to explicitly construct 2-colourings on exponentially sized graphs without large ...

**1**

vote

**1**answer

428 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 ...

**16**

votes

**0**answers

279 views

### Simpler proofs of certain Ramsey numbers

The reason for the gorgeous simplicity of the classic proofs of $R(3,3)$, $R(4,4)$, $R(3,4)$ and $R(3,5)$ is that essentially all you need is the trivial bound and a picture.
But for bigger Ramsey ...

**5**

votes

**2**answers

836 views

### Where can I find a catalog of known Ramsey numbers?

Is there an online catalog available of Ramsey numbers, preferably one that for unknown values documents the known upper/lower bounds?

**7**

votes

**1**answer

250 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 ...

**0**

votes

**0**answers

82 views

### A constrained minimum edge coloring

Is minimum number of colors needed to color edges of complete graph $K_n$ so that every even simple cycle contains at least one color assigned to odd number of edges at most $\beta n$ where ...

**2**

votes

**0**answers

74 views

### Even cycle constrained edge coloring

Is minimum colors needed to assign colors to edges of complete graph $K_n$ so that every $2t$ simple cycle where $t\in\Big\{1,\dots,2\Big\lfloor\frac{n}2\Big\rfloor\Big\}$ contains atleast $t+1$ ...

**10**

votes

**6**answers

1k views

### Algorithms for calculating R(5,5) and R(6,6)

Calculating the Ramsey numbers R(5,5) and R(6,6) is a notoriously difficult problem. Indeed Erdős once said:
Suppose aliens invade the earth and threaten to obliterate it in a year's time unless ...

**4**

votes

**1**answer

187 views

### Large bicliques in r-partite graphs containing no independent sets having one vertex from each class

Let $G$ be a multipartite graph on $r$ classes, each containing $k$ vertices, such that there is no independent set which contains at least one vertex from each class. I believe such graphs contain a ...

**10**

votes

**2**answers

618 views

### Does van der Waerden's Theorem hold for $\omega_1$?

One way to phrase van der Waerden's Theorem is:
For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is ...

**3**

votes

**1**answer

353 views

### Is this version of van der Waerden's Theorem consistent with ZFC?

One way to phrase van der Waerden's Theorem is:
For every finite coloring of $\mathbb N$ and every finite $F \subseteq \mathbb N$, there exist $a,b \in \mathbb N$ such that $a + b \cdot F$ is ...

**11**

votes

**2**answers

638 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 ...

**2**

votes

**0**answers

291 views

### What is the complexity of determining Ramsey Number?

In the notation of Garey and Johnson [1], two problems related to Ramsey Problem were defined:
$\textbf{ARROWING}$
Instance: (Finite) graphs $F$, $G$ and $H$.
Question: Does $F\rightarrow (G, H)$?
...

**28**

votes

**0**answers

715 views

### 3-colorings of the unit distance graph of $\Bbb R^3$

Let $\Gamma$ be the unit distance graph of $\Bbb R^3$: points $(x,y)$ form an edge if $|x,y|=1$.
Let $(A,B,C,D)$ be a unit side rhombus in the plane, with a transcendental diagonal, e.g. $A = ...

**6**

votes

**2**answers

128 views

### Reference request: monochromatic paths in edge-colored complete graphs

Given $k,c \in \mathbb{N}$, let $P(k,c)$ be the minimum $n$ such that no matter how we color the edges of the complete graph $K_n$ with $c$ colors, there is always a monochromatic path of length $k$.
...

**6**

votes

**0**answers

165 views

### A generalization of SOCA

Roughly speaking, SOCA (Semi Open Coloring Axiom) says that for an open coloring of the unordered pairs over an uncountable separable metric space you can always find an uncountable homogeneous subset ...

**1**

vote

**0**answers

51 views

### Genus tradeoffs in bipartite graph

Given $G$ as bipartite graph of genus $g(G)$ with number of vertices of each color being $N$ with $A$ as $N\times N$ biadjacency matrix. Denote $\bar{G}$ to bipartite graph of genus $g(\bar{G})$ of ...

**11**

votes

**3**answers

403 views

### Sets of points containing permutations - a Ramsey-type question

The following question arised as a side-question in a geometric problem. It has a "feel" similar to problems in Ramsey-theory, but I have not found any mention of it (also I'm not very familiar with ...

**17**

votes

**3**answers

579 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 ...

**11**

votes

**2**answers

739 views

### Where is the Erdős–Rado theorem stated in Erdős and Rado's Bull AMS paper?

This may be inappropriate for MO, but here goes: if I have understood the statement of the Erdős–Rado theorem correctly, then it contains as a special case the following result:
if $\mu$ is ...

**9**

votes

**1**answer

518 views

### 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 ...

**12**

votes

**2**answers

2k 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 $n>R(k, l)$, ...

**6**

votes

**0**answers

85 views

### 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 ...

**8**

votes

**3**answers

891 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 ...

**10**

votes

**2**answers

527 views

### 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 ...

**30**

votes

**4**answers

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 ...

**7**

votes

**2**answers

504 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

400 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 $s\in S$,
$$
...

**3**

votes

**1**answer

621 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 ...

**4**

votes

**2**answers

192 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 ...

**2**

votes

**0**answers

57 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.

**1**

vote

**0**answers

107 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

235 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?

**12**

votes

**6**answers

17k views

### 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

203 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 , ...

**16**

votes

**10**answers

3k views

### 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) ...

**5**

votes

**2**answers

747 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 ...

**2**

votes

**1**answer

317 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 ...

**3**

votes

**1**answer

127 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 $A_1,\dots,A_n$ ...

**1**

vote

**1**answer

121 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 ...

**8**

votes

**3**answers

1k 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 ...

**7**

votes

**3**answers

327 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 ...