Ramsey theory refers to questions of the form "how many objects are needed to guarantee that a given property of the collection holds?"

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A problem in prime number theory

I was wondering if anybody here might provide me with a hint for this rather innocuous-looking problem: If $X:= \{pq: p, q \mbox{ are prime numbers and } p\neq q\}.$ In addition, let us suppose that ...
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135 views

Ramsey number for paths

Let $n = R(P_{r+1}, c)$ be the smallest integer such that if $K_n$ is $c$-edge-coloured, then it contains a monochromatic subgraph isomorphic to $P_{r+1}$, the path of length $r$. I need to show that ...
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Does a red/blue coloring of the infinite subsets of $\mathbb{N}$ necessarily give an infinite monochromatic $M\subset \mathbb{N}$?

The infinite Ramsey theorem states that for any $n$, if all the subsets of $\mathbb{N}$ of size $n$ are colored red/blue, then there is an infinite $M$ all of whose subsets of size $n$ are ...
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183 views

How I can prove Ramsey number R(2,3,4) > 8?

I need to prove $R(2,3,4) > 8$ with Ramsey theory. How can I do that?
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75 views

Simple proof that W(3,3)=27?

I was wondering that if there exists a simple proof that the van der Warden's number W(3,3) (the smallest positive integer $n$ such that any 3-coloring of the set $\{1, 2, ..., n\}$ has a ...
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707 views

Good way to learn Ramsey Theory

What are some good books on Ramsey theory? I have Van Lints book on Combinatorics: is this enough preparation to start reading about Ramsey theory? I want a book that includes important results and ...
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How to prove some properties of partitions of finite subsets of N? [duplicate]

Possible Duplicate: Any partition of {1,2,..,9} must contain a 3-Term Arithmetic Progression The problem is as such: Prove that there is not a partition of $N_9 = \{1, 2, 3, 4, 5, 6, 7, ...
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237 views

Amalgamation of graphs

I am trying to understand the definition of amalgamation of a $n+1$-partite graph as explained here(first few lines of page 4). We have a $n+1$-partite graph $G$ with partite sets $V_0,V_1,\cdots,V_n$ ...
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98 views

Any forest on 5 or more vertices contains an independent set of size 3.

I am looking for a short proof of this fact. This is clearly true by drawing these trees, but I am having trouble putting it into writing. Somehow I need to select 3 of the 5 vertices and show that ...
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365 views

Monochromatic squares in a colored plane

Color every point in the real plane using the colors blue,yellow only. It can be shown that there exists a rectangle that has all vertices with the same color. Is it possible to show that there exists ...
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169 views

Any partition of $\{1,2,\ldots,9\}$ must contain a $3$-Term Arithmetic Progression

Prove that for any way of dividing the set $X=\{1,2,3,\dots,9\}$ into $2$ sets, there always exist at least one arithmetic progression of length $3$ in one of the two sets.
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Ramsey's theorem.

For $k=2$, from this post A "geometrical" representation for Ramsey's theorem, how one can deduce the theorem from Constant $f:[\mathbb{N}]^2\to \{1,2\}$ (part 2), or by knowing that ...
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Monochromatic triangle and edge colouring

$r(k) := R(\underbrace{3,3,...,3}_k)$ (I.e. $r(k)$ is the minimum integer $n > 0$ such that every coloring of edges of $K_n$ in $k$ colors is guaranteed to produce a monochromatic triangle.) Show ...
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444 views

Complete graph edge colouring in two colours: lower bound for number of monochromatic triangles

Say $K_n$ is a complete graph. Show that any coloring of edges of $K_n$ with $n \ge 6$ in two colors contains at least $$\frac1{20}\binom{n}3$$ monochromatic triangles. Any ideas on how to use Ramsey ...
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Coloring of positive integers

Suppose $f:\mathbb{Z}^+\longrightarrow X$ is a function, with $X$ a finite set. Is it true that there are $a,b\in\mathbb{Z}^+$ such that $f(a)=f(b)=f(a+b)$.
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What was done to calculate the Ramsey numbers using a quantum computer?

I recently came across this paper titled Experimental determination of Ramsey numbers with quantum annealing I was wondering what exactly the gist of the paper, as I read it, it seems rather ...
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Show that among 20 people there are either four mutual friends or four mutual enemies

Use the fact that among a group of 10 people, where any two people are either friends or enemies, there are either three mutual friends or four mutual enemies, and there are either three mutual ...
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225 views

Proof of Ramsey Theorem with explicit use of AC

What are minimal axiomatic requirements to prove Ramsey Theorem? This one: Let $X$ be some countably infinite set and colour the elements of $X^{(n)}$ (the subsets of $X$ of size $n$) in $c$ ...
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Prove Ramsey Number R(3,5)=14

I'm having problem proving the ramsey number of R(3,5) = 14. Below is my proof. Proof. Let $v_0$ be a vertex from a $k_{14}$ vertices. The vertices incident to $v_0$ are $v_1, v_2, \cdots , v_{13}$ ...
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van der Waerden's original proof

I am looking for a book/site which has the english translation of the proof of van der Waerden's theorem as presented by van der Waerden himself. In other words is the translation of the paper: ...
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1answer
178 views

Coloring Vertices of a 50-gon

This is a problem that I have spent a good 2 hours on but can seem to come up with any rigorous solution. If someone could provide one that would be great! If we color 21 vertices of a 50-gon, how do ...
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Ramsey (graph) theory question with tree and girth

Sorry for the abundance of questions I'm asking. Test is soon... Prove that for every tree $T$ and every $g \in \mathbb{N}$, exist $G$ with girth $g$, so that in any 2-edge-coloring of $G$ there is a ...
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137 views

Best known bounds for Ramsey numbers

I realize a similar question has been asked before but what I want to know is a little different and is not answered by the link in the answer to that question. I am interested in knowing the best ...
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79 views

2012-gon- subsets of vertices.

Can we prove or disprove this? For a sufficiently large $n$, every set of at least $ n$ points in the plane with no three collinear has a subset that form the vertices of a convex $2012$-gon. Gerry ...
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96 views

Complete graph-coloring

Can we prove or disprove the following statement? For any graph $H$ and any coloring $c$ of its edges with two colors, there exists $n$ such that every $2$-coloring of the edges of the complete graph ...
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Affine Van der Waerden's theorem

I have the following statement which is claimed to be a version of Van der Waerden's theorem: For any finite partition of $\mathbb{N}$, one of the cells contains affine images of every finite ...
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Graham's Number : Why so big?

Can someone give me an idea of how R.Graham reached Graham's Number as an upper bound on the solution of the related problem ? Thanks !
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Optimal sets for plane coloring problem

There is a reasonably well-known problem: Given a plane with each point colored one of k colors, show there is a rectangle whose vertices are all of the same color, whose axes are parallel to the ...
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181 views

Erdős Probabilistic method

My question is based on the Erdos probabilistic method. I am trying to read from the paper here. The proof of Theorem 1 contains the statement Since a block sequence is monochromatic with ...
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Monochromatic degenerate triangles in a two-coloring of the plane

In a similar vein to a question I asked a few days ago: Do all two-colorings of $\mathbb{R}^2$ contain three points of the same color which form the vertices of a degenerate triangle of side-lengths ...
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1answer
303 views

Gallai's theorem, colourings and equivalence relations

I'm revising a few past papers on Ramsey theory and I've come across a question which feels like it should be easy if it weren't so confusingly set up - I was hoping someone here could help me make ...
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Ramsey theory, finite colourings of $\mathbb{N}$ and infinite monochromatic sets

I am trying to show that the following statement is false: whenever $\mathbb{N}$ is finitely coloured by $c: \mathbb{N} \to \{1,\ldots,k\}$ (in the sense of Ramsey theory), there exists an infinite ...
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Ramsey Number R(4,4)

In trying to deduce the lower bound of the ramsey number R(4,4) I am following my book's hint and considering the graph with vertex set $\mathbb{Z}_{17}$ in which $\{i,j\}$ is colored red if and only ...
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Proof of the van der Waerden theorem

The van der Waerden theorem states that given any natural numbers $k$ and $r$, there exists a natural number $W=W(k,r)$ such that if the set $\{1,2\cdots W\}$ is divided into $r$ classes (also called ...
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Ramsey theory - colouring of edges

I'm trying to understand a proof: $R(3,3) = 6$ proof: Take a red/blue colouring of $K_6$. Take a vertex $v$ (is an element of) $V(K_6)$, either $v$ is incident to $\geq 3$ red edges or, $v$ is ...
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Prove the following inequality: $N(P,P,2)\leq 4^{P-1}$

I've made very little headway on this problem, so any help is appreciated. Edit: Sorry, I should have explained that. In general, $N(p,q,2)$ is the smallest value of $n$ such that a red-blue ...
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1answer
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Hales Jewett regularity theorem

I am trying to read the Hales-Jewett regularity theorem given as Theorem 1 here. I have a doubt in the proof which I am hoping someone here can clarify. Here are some background definitions and a ...
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Party problem / Ramsey's theorem R(3,3)

I'm looking for an algorithm that solve Party problem. The party problem asks to find the minimum number of guests that must be invited so that at least 3 will know each other or at least 3 will not ...
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Van Der Waerden without topological dynamics?

Applying topological dynamics to prove Van Der Waerden's theorem on the existence of monochromatic arithematic progression has now become a somewhat classical example of the power of topological ...
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Combinatorial Set Theory - Ramsey's Theorem Related Question

For a set $A \subseteq \omega$, let $[A]^n$ denote the set of subsets of $A$ of size $n$. I am trying to prove Ramsey's Theorem, and it seems like the following fact is used in the proof I am reading ...
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Understanding various definitions of TREE($n$) in Friedman's finite form of Kruskal's tree theorem.

I was reading the Wikipedia article on Friedman's finite form of Kruskal's tree theorem, and am interested in the large numbers TREE(n). I would like to verify TREE(2)=3 myself, but find conflicting ...
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Strengthened finite Ramsey theorem

I'm reading wikipedia article about Paris-Harrington theorem, which uses strengthened finite Ramsey theorem, which is stated as "For any positive integers $n, k, m$ we can find $N$ with the following ...
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Showing that $K_7$ contains at least 4 monochromatic triangles

A problem in my book is: Let the edges of $K_7$ be colored with the colors red and blue. Show that there are at least four subgraphs $K_3$ with all three edges the same color (monochromatic ...
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1answer
278 views

Algorithm to find a permutation that contains the fewest possible monotone subsequences of length $k$

Fix natural numbers $k,n$, with $k<n$. I want to find a permutation in $S_n$ that contains fewest monotone (increasing or decreasing) subsequences of length $k$. For example the permutation ...
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What is the proof of the “thin set theorem”? A result in infinite Ramsey theory.

OK so here's a precise question: Is it true that for every integer $k\geq1$ and every $f:\mathbb{Z}^k\to\mathbb{Z}$, there is some infinite subset $A\subseteq\mathbb{Z}$ such that $f(A^k)$ is not all ...
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Lower bound for monochromatic triangles in $K_n$

Say $K_n$ is a complete graph of $n$ nodes, and every edge is either blue or red. I'm trying to find $T_n$, which is the lower bound for the number of monochromatic triangles in $K_n$ (monochromatic ...
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How to prove this relation between Ramsey Numbers: $R(s, t) ≤ R(s, t-1) + R(s-1, t)$ for $s,t>2$

I am trying to prove that $$R(s, t) ≤ R(s, t-1) + R(s-1, t) $$ for $s,t>2$, where $R(s,t)$ is the Ramsey number of $(s,t)$, and I'd be really grateful for a hint that gets me started.
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Ramsey number for books

Given a triangular book $B_n$ I am trying to prove that $r(B_n,B_n)\le 4n+2$ where $r(B_n,B_n)$ is defined as the least positive number such that any graph $G$ on $r(B_n,B_n)$ vertices either has a ...
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1answer
158 views

Extended form of infinite Ramsey's Theorem

I'm working on some Combinatorics, and in the past have happily used the infinite Ramsey Theorem (as detailed on http://en.wikipedia.org/wiki/Ramsey%27s_theorem), which says that for something like ...
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An upper bound for a graph Ramsey number

I am trying to prove the following result, given as an exercise in my book: $r(K_m+\bar{K_n},K_p+\bar{K_q})\le\binom{m+p-1}{m}n+\binom{m+p-1}{p}q$. Here $r(G,H)$ denotes the Ramsey number for the ...