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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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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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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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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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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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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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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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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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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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 ...
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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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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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Ramsey Upper Bound improvement

I am trying to understand the upper bound that David Conlon produced for the diagonal Ramsey Numbers $$R(n+1,n+1) \leq n^{-c \frac { log n}{log \;{log n}}} \binom {2n}{n}$$ With the binomial theorem ...
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Conjectured Value of Ramsey Number $R(3,10)$

It is known that the value of the Ramsey number $R(3,10)$ is either 40, 41, or 42. Have any experts in the field offered a conjecture as to which it might be?
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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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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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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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Van der Waerden type numbers (for geometric progressions)

Van der Waerden theorem is true also for geometric progressions. Is there anything interesting in van der Waerden type numbers $ W'(r,k) $ derived from this version? ($ W'(r,k) $ is such that if the ...
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Questions on $R(K_l + \overline{K_t}, T_s) \le l(s-1)+t$

Following C. C. Rousseau and J. Sheehan "A class of ramsey problems involving trees" Journal of London Math. Soc. $1978$, I have some dudes about their proof for $R(K_l + \overline{K_t}, T_s) \le ...
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Van der Waerden number

The Van der Waerden number $w(l,k)$ is the least $n$ such that for every $k$-coloring of $[n]$ has a monochromatic $l$-term arithmetic progression. Prove that $w(l,k)>(lk^{l-1})^{1/2}$ Give some ...
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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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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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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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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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Ellis semigroup

I have only seen the construction of the Ellis semigroup applied to a group endowed with the discrete topology. Does the same idea works for any topological group? If not, where does it go wrong? (I ...
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Principal matrix, Ramsey theorem

Question: Let m be given. Show that if n is large enough, then every n-by-n 0, 1-matrix has a principal submatrix of size m in which all elements above the diagonal are the same, and all elements ...
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Equivalence of Theorems; the sphere on $\ell^2$ is finitely oscillation stable.

I just started reading the book Dynamics of Infinite-dimensional Groups, by Pestov, and right in the introduction the following theorem by Milman is cited: Let $\mathbb{S}^\infty$ denote the sphere in ...
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on binary strings without using Ramsey Theorem

Using the Ramsey Theorem Let $X$ be some countably infinite set and colour the elements of $X(n)$ (the subsets of $X$ of size $n$) in $c$ different colours. Then there exists some ...
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Lower Bound on VDW Numbers

I am trying to find the original proof given by E. Berlekamp that for prime $p$, $$W(p+1) \ge p2^p.$$ All the papers that I have searched only reference this result and give no proof.
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Algebra in Erdős's proof of lower bound for Ramsey number

A well known proof by Erdős shows a lower bound on the Ramsey number $r(k,k)$ using the probabilistic method. The theorem goes thusly: Let $n,\, k\in\mathbb{N}$ such that ${n \choose k}2^{1-{k ...
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Probability that a subset of a degree-regular graph shares at least a certain number of mutual connections

Consider a set of $n$ vertices of common degree $p$. What is the probability that some subset of $x$ vertices from $n$ share $q$ mutual connections within that group of size $x$? i.e. If we have ...
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Local Lemma on lower bounding R(k,k)

We aim to prove that if $k \ge 3$ and $e2^{1-\binom{k}{2}}\binom{n}{k-2} \le 1$ then $R(k,k) >n$ Now I understand that we colour the edges of $K_n$ red and black with probability 1/2. For each ...
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Link between Ramsey Theory, random graphs and spin glasses

As all three theories study the emergence of order, it would be natural if there were some links between: Ramsey theory; random graphs; spin glasses. Is there a textbook or an article that ...
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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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Ramsey number for two graphs

Given R(G, H) = n, we have the minimum number such that $K_n$ will always contain G in one color OR H in the other. Is this definition correct? I ask because I'm very confused about the particular ...
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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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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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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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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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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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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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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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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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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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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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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 lower bound 62?

Why is the lower bound of the minimum amount of points needed so that a $4$-coloring leaves at least one monochromatic triangle $62$, and not $66$? A lower bound of $66$ would seem obvious, since it ...
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Is $R(s+1,t) \geq R(s,t)$?

I've got a rather simple homework question in Ramsey theory that I'm struggling to answer. It is as follows: Is $R(s+1,t) \geq R(s,t)$ for all $s,t\geq2$? Is anyone able to lend a hand to get me ...