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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Why are there only a few known Ramsey numbers?

Can someone explain in a simple way, why there are so few known exact Ramsey Numbers? I guess it's because there are no efficient algorithms for this task, but are there so many combinations to test? ...
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Why König's lemma isn't “obvious”?

I keep facing König's lemma "Every finitely branching infinite tree over $\mathbb{N}$ has infinite branch". Why it is not taken "obvious" but needs a careful proof? It seems somewhat obvious, but I ...
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772 views

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 !
12
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457 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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How many 2-edge-colourings of $K_n$ are there?

I'm writing a paper on Ramsey Theory and it would be interesting and useful to know the number of essentially different 2-edge-colourings of $K_n$ there are. By that I mean the number of essentially ...
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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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Every 33-length subsequence of $1,2,\dotsc,122$ contains a three term arithmetic progression

Is it possible to prove that every 33-length subsequence of the sequence $1,2,3,\dotsc,122$ contains a three term arithmetic progression? Maybe I should post it on mathoverflow
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760 views

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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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 $v_1, v_2, \cdots , v_{13}$ ...
7
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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 ...
7
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234 views

Closest pair of vectors in $\{0,1\}^n$

Suppose we are given $k$ points in $\{0,1\}^n$ (using Hamming distance as metric). Consider the two points that have the smallest distance between them. Does there exist any results bounding this ...
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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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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 ...
6
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1answer
322 views

Known bounds and values for Ramsey Numbers

Is there a good online reference that lists known bounds on Ramsey numbers (and is relatively up to date)? The wikipedia page only has numbers for $R_2(n,m)$. I am specifically interested in known ...
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2answers
126 views

Two subsets and their union have same color

Color all nonempty subsets of $[n] = \{1,2,\ldots,n\}$ with colors $1,2,\ldots,r$. Show that, for all large enough $n$, there exist two disjoint nonempty subsets $A,B \subseteq [n]$ such that ...
6
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195 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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What is the explanation for the $64$ in Graham's number $g_{64}$?

As in, why does the iteration of the function until $g_{64}$ guarantee this property that defines Graham's number? Why was this number chosen? If I had to guess (emphasis on guess), I'd say that the ...
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2answers
551 views

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 ...
6
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196 views

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 ...
6
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1answer
625 views

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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Prove that $r(k,k) + k \leq r(k + 1, k + 1) $

Prove that $r(k,k) + k \leq r(k + 1, k + 1)$, where $r(k,l)$ is the minimum number of vertexes in a Graph, where we have a clique with $k$ vertexes or a stable set with $l$ vertexes. There are ...
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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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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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How long does a sequence need to be to be guaranteed to have a monotonic subsequence length k?

The sequence 7, 2, 4, 1, 4, 8 has an increasing subsequence length four (2, 4, 4, 8) and a decreasing subsequence length three (7, 4, 1). It has other monotonic (increasing or decreasing) subsequences ...
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Old exam question about well ordering on $\mathbb{R}$

I am not sure how to start tackling this question and would love a hint: Let $<$ the regular order relation on $\mathbb{R}$, and $<_w$ well ordering on $\mathbb{R}$. We define a coloring ...
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A question about infinities and pots of paint

This question is inspired by http://math.stackexchange.com/a/1052384/66307 and quotes from it heavily. Take a countably infinite paint box; this means that it has one color of paint for each positive ...
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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 ...
5
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1answer
267 views

What's the difference between Ramsey theory and Extremal graph theory?

Wikipedia teaches us that problems in Ramsey theory typically ask a question of the form: "how many elements of some structure must there be to guarantee that a particular property will hold?" It ...
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1answer
305 views

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 ...
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Partitioning Integers into Equal Sets to Guarantee Arithmetic Progression

I've run into the following problem which I am sure is true but I cannot prove it: If we color the integers in the set $S = \{1, 2, \ldots, 3n \}$ with $3$ colors such that each color is used $n$ ...
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number of potential couples

A potential couple is a pair of a man and a woman that like each other (assume that 'like' is a symmetric relation). Given a group of $M$ men and $W$ women, I want to know how many different ...
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Does “Big Data” Have a Ramsey Theory Problem?

I'm erring on the side of conservatism asking here rather than MO, as it is possible this is a complex question. "Big Data" is the Silicon Valley term for the issues surrounding the huge amounts of ...
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Counterexample for $R(4,4) \neq 8$

I try to find a counterexample for $R(4,4)\neq 8$. (R is the Ramsey-number). I drew a graph with 8 vedges and I coloured all edges $(v_i,v_j)$ with $i-j =\pm 2,4,6$ in the same colour (for example ...
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There exist $\{a_{n}\},\{b_{n}\}$ such $\lim_{n\to\infty}\frac{a_{n}}{b_{n}}=c?$

Let $ A$ and $ B$ are two infinite subsets of the natural numbers $\mathbb{N}$, such that $$A\cap B=\emptyset \qquad A\cup B=\mathbb N$$ Question: is it true that for every natural $c>0$, ...
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What is Ramsey Theory ? what is its own importance in maths?

3 days ago , i had a discussion with a close friend who studies physics - still a student - . and i was telling her about the biggest known numbers in maths , so i told her about numbers such googol ...
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Guaranteeing an integer lattice point centroid

My question is this: Writing $n(4)$ to be the minimum number of integer lattice points in the plane so that some four of them must determine an integer lattice point centroid, show that $n(4)=13$. I ...
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Crowding the boundary of non-constructivity without crossing it?

Can any sense be made out of my vague feeling that some proofs in Ramsey theory are as close as you can get to non-constructive proofs without crossing the line? Is there any way to make this ...
4
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Given an infinite poset, show that it contains either a infinite chain or an infinite totally unordered set.

Been thinking about this one for awhile and I'm still stuck... Thought about proving it by arriving at a contradiction but I haven't reached anything noteworthy. If you do a proof by contradiction, ...
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how to construct 17-vertexed graph for Ramsey number R(3,6)=18

Ramsey number $R(3,6)=18$. How to construct a graph of $17$ nodes which does not contain neither a clique of order $3$ or an independent set of order $6$. could you show me the tactics or the ...
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A Ramsey-type result for families of subsets

Let $S$ be a set of cardinality $\aleph_1$. Consider the directed family $\mathcal{C}$ (here directed means directed with respect to the inclusion) of all countably infinite subsets of $S$. Suppose ...
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1answer
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$4$-cycle of the same color in $K_n$

Let $k$ be a fixed positive integer. All edges of the complete graph $K_n$ are colored in one of $k$ colors. What is the least $n$ such that there always exists a $4$-cycle of the same color? This ...
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1answer
368 views

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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Edge coloring a graph to find a monochromatic $K_{2,n}$

I am trying to prove or disprove the following statement: Let $n>1$ be a positive integer. Then there exists a graph $G$ of size 4n-1 such that if the edges of $G$ are colored red or blue, no ...
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Arithmetic progression in a subset of $\mathbb N$

What non-trivial sufficient and/or necessary conditions are there for existence an arithmetic progression (finite or infinite length) in an infinite subset of $\mathbb N$.
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1answer
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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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Euclidean Ramsey theory problem

Let $k\geq 1$ be given. Consider the following statement: For all (non equilateral) triangles (represented by 3 points in $\mathbb R^2$) and for all $k$-colorings of $\mathbb R^2$ there exists a ...
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1answer
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Do Ramsey idempotent ultrafilters exist?

I was studying idempotent ultrafilters when I saw that no principal ultrafilter could ever be idempotent, because $\left\langle n \right\rangle \oplus \left\langle n \right\rangle = \left\langle 2n ...
4
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1answer
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The Erdős-Szekeres problem on points in convex position

The Erdős-Szekeres problem on points in convex position and its proof using Ramsey theorem are well know. The problem goes like this: For every natural number $k$ there exists a number $n(k)$ such ...
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History of Hindman's Theorem

At this blogpost about Hindman's Theorem, I read the following lines: 'I love the odd history so allow me to digress... etc. ' This sentence made me curious to know what this history looks ...
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Arbitrary ratio sequences on a partition of $\mathbb{R}$ (Partition regularity of fixed ratio sequences)

Background: This question arose purely recreationally and doesn't really fit into any context that I know of. Let $A \sqcup B = \mathbb{R}$ be a partition of the ...