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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Find maximal clique in an multigraph with $n$ vertices, where each vertex is colored with $k$ colors.

You are given a multigraph with $n$ vertices. Every vertex is colored with maximum of $k$ colors. If two vertices share a color, there is an edge between them which is colored with that color. (A pair ...
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A Ramsey not Completely Ramsey Set of $[\omega]^\omega$.

Let $a \in [\omega]^{<\omega}$ (a finite subset), $A \in [\omega]^{\omega}$ (an infinite subset). Let us define $$[a, A] = \{a \cup B: B \in [A]^{\omega} \wedge max(a) < min(B) \}.$$ These ...
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Aplication of Ramsey theory in group theory

Let $G$ be a infinite group. How to show that there is a $A\subseteq G$ infite such that $$\forall x,y,z\in A\;\; \big(xy=z\Leftrightarrow (x=y=z=x^2)\big)$$ I've tried to define the coloring $c:[...
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Asymptotic lower bound for R(k,k)

I'm reading Spencer's lectures on the probabilistic method. Using the Lovasz local lemma, we've shown that $R(k,k)>n$ if $$ 4{k \choose 2} {n \choose k-2} 2^{1-{k \choose 2}} < 1. $$ Now I'm ...
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Number of edges needed for good colouring in Ramsey graph theory

Given $n$, consider the complete graph $K_{R(n)-1}$, where $R(n)$ is the diagonal Ramsey number. So there exist $2$-colourings of the edges of $K_{R(n)-1}$ without a monochromatic copy of $K_n$. ...
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Show that in any group of 9 people there is always a subgroup of 3 mutual strangers or a subgroup of 4 mutual acquaintances.

Show that in any group of 9 people there is always a subgroup of 3 mutual strangers or a subgroup of 4 mutual acquaintances. I know that this is an application of Ramsey's Theorem, but I'm not sure ...
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113 views

Pigeon-Hole Principle Common Sum

Each of 15 red balls and 15 green balls is marked with an integer between 1 and 100 inclusive; no integer appears on more than one ball. The value of a pair of balls is the sum of the numbers on the ...
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Problem on Ramsey Numbers

I want to prove that $R(3,5) = 14$. I think the easiest first step is proving $R(3,5) \leq 14$ since I think developing a counterexample on 13 vertices would be cumbersome. Let's say the two edge ...
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$n$ distinct real numbers has a monotone subsequence of length $k$ if $n \ge (k-1)^2 + 1$

I'm working through some problems and I just completed the proof that if $n\ge R(k)$ for a sequence of distinct real numbers $a_1, a_2, a_3, ..., a_n$ has a monotone subsequence of length $k$, that is ...
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What are some important applications of the Erdős-Szekeres Theorem?

Erdős-Szekeres Theorem: Any finite sequence of $n^2+1$ real numbers contains a monotonic subsequence of length at least $n+1$. I was wondering what are the most important applications of the Erdős-...
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Prove that there is a red triangle or a blue triangle that is is a sub-graph

If the edges of $K_6$ are coloured blue or red, prove that there is a red triangle or a blue triangle that is a sub-graph. Well I am having a hard time proving this, I try to prove it by ...
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How to create a matrix in Hammersley's proof for Erdős-Szekeres Theorem?

Hammersley gave the following algorithm that proves the theorem. Let a sequence $a_1,a_2,...,a_{n^2+1}$. (a) let $a_1$ start the first column and for $i\ge 1$ (b) if $a_i$ is greater than or equal ...
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What does Ramsey theory tell us?

I have recently started reading about Ramsey theory, though I'm a bit confused about what does it actually tell us. As long as I understood, it says that in a big enough complete graph one can find a ...
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Erdős-Szekeres theorem generalized example showing exactness

I am struggling to understand the following example taken from Seidenberg's paper (1959). "A well-known example of a sequence of $mn$ terms like the following: $$m,m-1,\ldots,1,2m,2m-1,\ldots,m+1,3m,...
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Ramsey coloring of $K_{13}$

Arrange the vertices of $K_{13}$ in such a way that they form a regular $13$-gon. Color the edges (which are now either edges or diagonals of the 13-gon) in read and blue, where an edge is colored red ...
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140 views

Ramsey number inequality proof [duplicate]

Let $2 \leq p \leq q$ and $2 \leq r \leq s$. Prove that $R(p,r) \leq R(q,s)$ and that equality holds if and only if $p=q$ and $r=s$. The equality part is clear, cause we will have $R(p,r) = R(p,r)$, ...
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Use of pigeonhole principle in ramsey-theorem about monochromatic triangles.

Im trying to prove that for any number n the complete graph with $p(n)$ vertices whose edges have been colored with n colors in some way has a monochromatic triangle (a triplet of nodes that are ...
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An example showing that van der Waerden's theorem is not true for infinite arithmetic progressions

One of the possible formulations of Van der Waerden's theorem is the following: If $\mathbb N=A_1\cup \dots\cup A_k$ is a partition of the set $\mathbb N$, then one of the sets $A_1,\dots,A_k$ ...
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Are some results about coloring positive integers valid for other semigroups?

There are some results in Ramsey theory, which involve additive structure of $(\mathbb N,+)$. For example, if we color the set $\mathbb N$ by finitely many colors, then: There are three numbers $x$,...
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Lower bound for $R(3, 3,\ldots, 3)$

As part of learning Ramsey numbers I am trying to prove that $R(\underbrace{3, 3,\ldots, 3}_{k\text{ times}}) > 2^k$ using the constructive method. In order to do that one needs to colour the edges ...
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72 views

Lower bound for the Ramsey number $r(k,k)$

I'm trying to prove the following inequality for every natural $k$: $$r(k,k)>(k-1)^2$$ I was trying to find a blue-red edge coloring of $K_{(k-1)^2}$ without either red or blue $K_k$. Any ideas?...
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The proof of Ramsey's Theorem

I try to understand the proof of Ramsey's Theorem for the two color case. There are still some ambiguities. It says $R(r-1,s)$ and $R(r,s-1)$ exists by the inductive hypothesis. I know the principle ...
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An upper bound on van der Waerden Numbers W(r, k), determined from the Number of Colorings r

Let $W(r, k)$ be a van der Waerden number, such that the interval $[1, W(r, k)]$ contains an arithmetic progression (AP) of $k$ terms, (k > 1), where the integers in the AP all have the same (...
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Find the largest possible value of $n$: color segments connecting any 4 of $n$ points with 4 colors

Let $A_1, A_2, \dots, A_n$ be $n$ points on the plane, no three collinear. Each of the segments connecting two points are colored by one of four given colors. Find the largest natural number $n$ ...
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42 views

How injective must these functions be?

Let $S$ be a finite set, and let $b:S\rightarrow\mathbb{N}$ be a function (note, $\mathbb{N}$ includes $0$). For any $K\subset S$ and a function $p:K\rightarrow\mathbb{N}$, let $P_{b,p}$ be the set of ...
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Why the Ramsey number $R(2,4)$ is not equal to $2$?

I'm reading Harris/Hirst/Mossinghoff's: Combinatorics and Graph Theory. Here: I don't understand: For all $2$-colorings, it must have a $K_p$ and $K_q$ or it must have a $K_p$ or a $K_q$? I'm ...
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Congruent quadrilaterals in a tri-colored $72$-gon

I recently watched a movie (A Brilliant Young Mind) in which this problem appeared: Let the vertices of a regular $72$-gon be colored red, blue, and green in equal parts. Show that there are $4$ ...
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Upper limit on Ramsey number $R(a,b)$

How could we prove that if $R(a-1,b)$ and $R(a,b-1)$ are both even then $R(a,b)$ is strictly less than $R(a-1,b)+R(a,b-1)$ or $\begin{equation} R(a,b) < R(a-1,b)+R(a,b-1) \end{equation}$
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Salem Spencer Theorem

The Salem Spencer Theorem seems to be a very interesting combinatorial theorem. This blog motivated me to read more about it. I understand the statement of the theorem, however the proof isn't very ...
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Reference for Ramsey Numbers

Just wondering about diagonal Ramsey numbers $R(n)$. Can anyone provide reference on either of the following? Have there been any notable attempts to make sense of $R(n)$ by using non-combinatorial ...
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Ramsey number $R(K_4,K_4,K_4)$.

I've done a bit of googling, but I can't seem to locate any bounds for $R(4,4,4)$. Here, $R(n_1,n_2,n_3)$ is the generalized Ramsey number where $n_1,n_2,n_3$ are orders of complete graphs. So, in ...
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A question on arithmetic progressions

Is it true that for every $n \in \mathbb N$ , $\exists N \in \mathbb N$ such that for any subset $A \subseteq \{1,2,...,N\}$ , either $A$ or $\{1,2,..,N\} \setminus A$ contains an arithmetic ...
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tuple of integers

The integers 1,2,...,30 are invited to a dinner party. They all sit around a round table, in some unknown order. Does there exist an ordering in which there are no three successive (successive means ...
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Brain teaser solution in Graph Theory / Ramsey Theory

I have a solution to the following brainteaser, which I think is the correct answer, but I haven't been able to come up with a way to prove that it's the right answer. I know very little about graph ...
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Calulating the Ramsey number $R(T, K_{1,n})$ of a tree $T$ and bipartite graph $K_{1,n}$

Let $m,n \ge 2$ be such that $m-1$ is a divisor of $n-1$. Let $T$ be a tree with $m$ vertices. Calculate the Ramsey number $R(T,K_{1,n})$. Thoughts: I'm having trouble approaching this question. I ...
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Another Evaluation of the Ramsey number $\mathcal{R}(3,3,3)$

The problem Show that $\mathcal{R}(3,3,3)=17$ The story behind the problem and some notation It was first proven by Greenwood and Gleason in 1955 in their paper Combinatorial relations and ...
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Is this equivalent to Szemerédi's theorem?

I know that Szemerédi's theorem states that any set of integers with positive natural density contains arbitrary long arithmetic progressions. However, does this imply that such a set contains an ...
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Graph: What is $R(K_{1,5},K_{1,5})$.

We define $R(H_1,H_2)$ to be the least number such for every graph $G$ with at least $R(H_1,H_2)$ vertices, either $H_1\subset G$, or $H_2\subset G^c$. What is $R(K_{1,5},K_{1,5})$ ? I would say that ...
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Why is the Ramsey`s theorem a generalization of the Pigeonhole principle

German Wikipedia states that the Ramsey`s theorem is a generalization of the Pigeonhole principle source But does not say why this is true. I am doing a presentation about the Ramsey theory and also ...
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Canonical colorings over $ \omega $

Given a natural number n, let $ c:[X]^n \to \omega $ be a coloring by arbitrary many colors, where $X$ is an infinite countable set. Then there exists an infinite subset $ H $ of $ X $ for which the ...
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Finding higher Ramsey numbers

How do mathematics go about finding larger Ramsey numbers such as R(5, 5)? How do they find upper bounds on these numbers?
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Proving $R(3,4)\le 9$

I am trying to prove $R(3,4)\le 9$. This is my approach: For any $K_9$ we have (WLOG) at least 4 red edges by the pigeonhole principle. Consider all of the edges between these 4 red edges, if ...
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What is $R(k,l)$?

I'm reading Landman/Robertson's: Ramsey Theory on the Integers. It states the following theorem: Theorem 1.15 (Ramsey's Theorem for Two Colors). Let $k,l \geq 2$. There exists a least positive ...
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A math contest question related to Ramsey numbers

In a group of 17 nations, any two nations are either mutual friends, mutual enemies, or neutral to each other. Show that there is a subgroup of 3 or more nations such that any two nations in 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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370 views

Rectangular stained glass window with different colors

Suppose you have six squares of stained glass, all of different colors, and you would like to make a rectangular stained glass window in the shape of a 2 × 3 grid. How many different ways can you do ...
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Prove that any group of 14 people must contain either 5 mutual friends or 3 mutual strangers.

So I think I have the answer to this problem, but there's something about it that's bothering me: Suppose we choose a fixed point with $13$ edges coming out of it. There must be at least $a)$ $9$ ...
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Conjectured diagonal Ramsey numbers

While doing some reading on the Wikipedia page for Ramsey numbers, I stumbled upon OEIS sequence A120414, which lists the allegedly conjectured values of the diagonal Ramsey numbers $R(n,n)$. I ...
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How to computably reduce the number of colors in (infinite) Ramsey's theorem

Suppose we have an "oracle" that gives a homogeneous set for a 2-coloring $\hat c : [\omega]^2 \rightarrow 2$ of pairs of integers. Using this oracle, can we "compute" a homogeneous set for a 3-...
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Ramsey numbers: if $s_1 \leq s_2$ then $R(s_1,t)\leq R(s_2,t)$

I'm doing this little homework assignment on Ramsey numbers, the question is: Show that $$s_1 \leq s_2 \Rightarrow R(s_1,t)\leq R(s_2,t).$$ I've tried classifying it into these four cases: The ...