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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Prove that there exists a one-color $K_3$ in a $K_{17}$ which is colored with three colors

Assume that we have a $K_{17}$ and we color every edge of it with 3 colors ( Like Red, Blue & green ). Prove that for every coloring of $K_{17}$ with 3 colors, After coloring, We have a ...
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Prove that every edge-coloring of $K_{17}$ with $3$ colors contains a monochromatic $K_3$. [duplicate]

Also, Prove that every edge-coloring of $K_6$ with $2$ colors contains at least two monochromatic copies of $K_3.$ I have no idea how to start these problems. What should I do?
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Colouring $K_{2s-1}$

Suppose we 2-colour $K_{2s-1}$ such that no vertex has more than one blue edge incident to it, prove that the graph contains a red $K_s$. I've never seen a Ramsey theory question like this and am ...
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35 views

Total number of gifts given at the end of a party

The following is true for n guests at a Christmas party: ...
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1answer
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Graph Theory Proof that R(3,4)=9

The attatched is supposed to prove that $R(3,4)=9$ . One line say says there is no red $K_3$ in the two-colouring of $K_8$ What is it talking about?- I can see plenty of red triangles! (with corners ...
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37 views

Ramsey Number algorithm

In general is there an algorithm to obtain Ramsey Number? For example how should I approach to get $R(2K_2, 2K_2)$ or R(3, 2K_2)?
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Coloring classes of $\{1,2,3,\dots,n\}$

I'm trying to prove the following statement There is an integer $n_0$ such that for any $n\ge n_0$, in every $9$-coloring of $\{1,2,3,\dots,n\}$, one of the $9$ color classes contains $4$ integers ...
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expressability of finite and infinite ramsey theorems in Peano arithmetic

finite ramsey theorem: for all e,k,r natural numbers, there exists a least natural number m=R(e,r,k) so that for all sets M when cardinality of M is larger or equal m and all of the e- tuples of M are ...
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38 views

Ramsey theorems for the naturals and for general infinite sets

In reverse mathematics and in recursion theory, the infinite Ramsey theorems are usually stated in terms of coloring of $[\Bbb N]^n$. How do these (not) imply the Ramsey theorems for general infinite ...
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Sufficiently many points in $\mathbb{R}^d$ must contain $m$ points forming the vertices of a convex polytope?

Let us say that a set of points in $\mathbb{R}^d$ is minimal if it forms exactly the set of vertices of a convex polytope. Equivalently, no proper subset of the points has the same convex hull; no ...
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Graph or its Complement contains a triangle.

How do I prove that the graph with at least 6 vertices or its complement contains a triangle? Do I have to prove that if a graph contains a triangle, then its complement doesn't contain, and the ...
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27 views

Prove that if $n > 1$ such that $n \to (a,b)$ and $m > n$, then $m \to (a,b)$.

I've simplified the question's notation to read as follows, If $n > 1$ such that every graph $H$ on $n$ vertices has $\alpha(H) \ge a$ or $\omega(H) \ge b$ and $m > n$, then every graph $G$ ...
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22 views

Find a graph with 5 vertices such that $\omega(G) = 2$ and $\alpha(G) = 2$.

I am trying to show that the following statement is false by providing a drawn graph as a counterexample. I've been pointed in the right direction that the statement is only true for at least 6 ...
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1answer
87 views

How do I get this upper bound for Ramsey numbers: $R_k \le \left \lfloor k!e \right \rfloor + 1$?

For every integer $k \ge 2$, $$R_k \le \left \lfloor k!e \right \rfloor + 1$$ where $R_k$ denotes $R(\underbrace{{3, 3, \ldots, 3}}_{k})$.
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What is the smallest $n$-uniform transversal of $\binom{[2n+2]}{2n}$?

For a family of sets $F \subseteq 2^{[m]}$, let $T \subseteq \binom{[m]}{n}$ an $n$-uniform transversal of $F$ if and only if $\forall f \in F~\exists t \in T: t \subseteq f$. In other words, each ...
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Let $n \ge 6$. Prove that it is not possible to partition the edges of $K_n$ into floor($\frac{n}{6}$) planar subgraphs. [duplicate]

Any hints will be appreciated. Here are some things I thought about: Since $K_n$ is a complete graph, it must be $(n - 1)$ regular The sum of the degree of all its vertices is $n(n - 1)$ There are ...
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Prove that the Ramsey number $R(4,4;3) = 13$

Prove that the Ramsey number $R(4,4;3) = 13$. I don't know how to deal with the Ramsey number $R(p_1,p_2,...,p_k;r)$ where r is larger than 2. Is there any useful inequality or construction of ...
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An extremal combinatorics problem

Given $n\in\Bbb N$, $\alpha\geq1$ denote $f(n,\alpha)$ as worst case minimum number of columns among all $n\times n^\alpha$ $0/1$ matrices with every row summing to $>\frac{n^\alpha}2$ that is ...
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47 views

Coloring Equilateral Triangles in $\mathbb R^n$

We start with this Example: No matter in which way you color the points of $\mathbb R^4$ with two colors, you can always find an equilateral triangle with vertices of the same color. In fact in ...
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What's the deal with Ramsey graph theory with vertex-colourings instead of edge-colourings?

As in the edge-colouring case, we can talk of a $r$-Ramsey graph $R$ for some (finite) graph $G$ wrt. vertex-colouring, i.e. such that for every $r$-colouring of the vertices of $R$ there is a copy of ...
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A combinatorial proof for a bound on diagonal Ramsey numbers

I wish to prove $R(p,p)\leq\frac{2^{2p-2}}{\sqrt{p}}$ combinatorially. I have proved this algebraically through the definition of the binomial coefficient but I would much prefer a proof from ...
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34 views

Prove that a function described as below exists

$R(n,k,l)$ is defined like this : Imagine we have a set and we want to color every subset of it having $k$ elements with $n$ colors such that at the end of coloring, there exists a subset with $l$ ...
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Approaching Ramsey theory in a unified way

Ramsey theory consists of seemingly diverse results like van der Waerden theorem and Ramsey theorem. There does not seem any apparent connection between these beyond the usual "Order within disorder" ...
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prove that $R^*(r+1,k;a_1,…,a_r,k)=R^*(r,k;a_1,…,a_r)$

Imagine we have a set with $n$ members. we want to color $k-subsets$ of this set with $r$ colors called $c_1,\ldots,c_r$ such that one of these things happen : - we have a set with $a_1$ members such ...
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50 views

an edge coloring of $k_{16}$ with no monochromatic triangle [closed]

My plan is to show that $R(3,3,3)$ is more than 16. So, i want to prove it with graph-theory. i know i should find an edge coloring of $k_{16}$ which contains no monochromatic triangles. Can anyone ...
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22 views

can we find a $k_4$ colored with 1 color in a $k_8$ which is colored with just 2 colors?

Assume that we define edge coloring in this way : An edge coloring of a graph is an assignment of "colors" to the edges of the graph. So, now imagine we have a $K_8$ which has edges colored with just ...
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1answer
66 views

Uses of Ramsey Theory in Astronomy?

In the last paragraph of a Scientific American article of July 1990 that can be found here http://www.math.ucsd.edu/~ronspubs/90_06_ramsey_theory.pdf Graham and Spencer write "Today we can easily ...
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62 views

Ramsey counter examples

I do not know of any solution or if it's an open problem: Let $R(i,i)=k$, therefore there exists a counter examples with blue and red edges for a clique of size $k-1$. Does there exist a ...
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Monochromatic triangle in two closed set which cover the plane

I am reading Section on Euclidean Ramsey Theory in Ronald Graham's Rudiments of Ramsey Theory. Exercise 7.3 states that Show that if $E^2$ is covered by two closed sets of colors then every ...
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in every coloring $1,…,n$ there are distinct integers $a,b,c,d$ such that $a+b+c=d$

Prove that for every $k$ there is a finite integer $n = n(k)$ so that for any coloring of the integers $1, 2, . . . , n$ by $k$ colors there are distinct integers $a, b, c$ and $d$ of the same color ...
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79 views

Lower bound for Ramsey numbers: $R( n + 2 , 3 )>3n$

I need to prove the following inequality: $$R( n + 2 , 3 )>3n$$ where $n>1$ and $R(s,t)$ is a Ramsey number. The most general way to prove such inequalities is to paint a graph with $3n$ ...
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71 views

Infinitely many points in plane s.t. no point is a convex combination of other points

Let $A$ be an infinite set of points in the plane, with no three points of $A$ collinear. I want to prove that $A$ contains an infinite set $B$ such that no point of $B$ is a convex combination of ...
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Ramsey number for tree and complete graph [duplicate]

I am having a lot of trouble understanding Ramsey theory. I am working on an exercise that asks for the Ramsey number $R(T,K_{1,n+1})$ where $T$ is a tree with $m$ edges and $n$ is a multiple of $m$. ...
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Independent Set of Product Graph and Ramsey Number

For two graphs $G,H$, define $G\otimes H$: it has vertex $V(G)\times V(H)$, $(v_1,v_2)(v_1',v_2')\in E(G\otimes H)$ if it satisfies all the following three requirments (i) $(v_1,v_2)\neq (v_1',v_2')$ ...
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Maximum $K_6$-free graph on $15$ vertices?

My question is, suppose we have $15$ vertices. What is the maximum number of edges we can add between these vertices, each with a fixed colored red or blue, so that there is no monochromatic triangle? ...
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Finding the exact value for $H(7)$

The graphs that I work with are all complete, each edge is colored red or blue, and each vertex is colored red or blue. $\textbf{Definition:}$ A graph is $\textit{Happy}$ if there exists a vertex ...
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94 views

Proof involving Ramsey numbers

$S$ is a set of R(m,m;3) points in the plane in which no 3 points are collinear. I am trying to prove that $S$ contains $m$ points that form a convex $m$-gon. I have tried using similar logic to the ...
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Finite coloring of $[\omega]^{<\omega}$.

It is known that Ramsey theorem does not hold for finite colorings of $[\omega]^{<\omega}$. So I am interested in this "partial" result: First let $S_n = ]n, +\infty[$ be the set of natural ...
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49 views

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 ...
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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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93 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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41 views

$n$ 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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101 views

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 ...
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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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42 views

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 ...