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Hereditary and Pre-compact implies Compact

Lemma II.3.20 in the book Ramsey Methods in Analysis by Argyros and Todorcevic states that a pre-compact hereditary family $\mathcal{F}$ of finite subsets of $\mathbb{N}$ is compact. It is supposed to ...
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1answer
81 views

Mixed Tsirelson Norm

The following is a definition of a Banach space that is a generalization of the original Tsirelson space. Nowadays such a space is called a Mixed Tsirelson space; it was introduced by Argyros and ...
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0answers
45 views

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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1answer
16 views

Give an explicit 2-coloring of the edges of Kn that proves R(k,k)> (k-1)^2

Right now I have: the coloring that there are k-1 subgroups of k-1 vertices. If each of the subgroups contains a connected graph that's one color (like black), and the edges between the subgroups is ...
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3answers
50 views

Is there only one counter example in $K_5$ for $R(3,3)$?

Title says it all. And the one that I know is below. (image from wikipedia) My question is: Is there only one counter example in $K_5$ for $R(3,3)$ where $K_5$ is a complete graph of 5 points and ...
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1answer
62 views

Homework question about Ramsey numbers

Consider a group of nine people. We know that at least one person, say Adam, knows an even number of people and does not know an even number of people. Show that either Adam and two other people all ...
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1answer
46 views

Combinatorial Question using ramsey's theory or pigeonhole principle??

We are currently going over pigeonhole principle, ramsey's theorem (graphs and such). Stuck on this particular question: Within a group of an odd number of people, show that at least one person knows ...
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3answers
28 views

Prove: For any 2 coloring of 2-space, one of the color classes contains points at all distances

We color the 2D-plane either red or blue at every point. Prove that one of the sets (either red or blue), contains two points at distance $D$, for every positive real number $D$.
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2answers
40 views

Coloring of $K_{17}$

For any 3-coloring of $K_{17}$ I have to show there exists either a red, blue or green triangle. To start, can I use proof by contradiction with color red, blue, green? So $(0,0,136)$ means all 136 ...
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1answer
60 views

Two coloring questions and ramsays number

What is the smallest $n$ such that every 2-coloring of edges of $K_n$ contains a red or blue 4-cycle (not $K_4$)? I am given that $R(4,4) \le 18$ and $R(3,5) \le 14$ Any help is greatly appreciated!
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2answers
88 views

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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1answer
39 views

Uncountable monochromatic set

Maybe you can help me with that. I was asking myself if you take an uncountable set $S$ and let $S^{(2)}$ be 2-coloured, must there exist an uncountable monochromatic set in $S$?
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1answer
29 views

Special Ramsey number $r(G)$

With $r(G)$ I refer to the smallest $n$ such that every blue-red colouring of the edges of $K_n$ contains a monochromatic copy of the grpah $G$ (this exists because $r(G)\le R(|G|)$). Now let $I_k$ ...
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1answer
11 views

Using Ramsey theory to show some properties of subgraph of a directed graph

Let $r > 1$ be an integer. Prove that there is an integer $n_0$ such that for every integer $n\geq n_0$ and every directed graph $G = (\{1,...,n\}, E)$ without loops, $G$ has an induced ...
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1answer
90 views

Ramsey Number for Star graphs

For two graphs $H_1$ and H2, the Ramsey number $r(H_1, H_2)$ is the minimum number r so that in any red-blue coloring of the edges of the complete graph Kr on r vertices there is necessarily either a ...
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59 views

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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4answers
127 views

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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1answer
51 views

Monochromatic solutions to $x_{1}+x_{2}+\cdots+ x_{m}=x_{m+1}$

What is the least $n$ such that any $2$-coloring of $[n]$ has a monochromatic solution to the equation $$x_{1}+x_{2}+\cdots+ x_{m}=x_{m+1}.$$ I think the answer is $m^{2}+m-1$. I have a coloring of ...
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1answer
121 views

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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3answers
69 views

Why is this binomial coefficient bounded thus?

Source: Miklos Bona, A Walk Through Combinatorics. $$ \forall k\geq 2,\binom{2k-2}{k-1}\leq4^{k-1}.$$ The RHS is the upper bound of the Ramsey number $R(k,k)$. How can I prove the inequality ...
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2answers
48 views

$k$ colorings of the non empty subsets of $[n]$ gives the same color to two disjoint sets and their union.

This question was already asked but I didn't get enough information from the answer. Here is a link to the question. Here is the question restated. Show that for $n$ large enough, every $k$ coloring ...
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1answer
149 views

Schur's theorem and infinite version

I've got a homework exercise on Schur's theorem, which says that for any $r \in \mathbb N$ there is an $n \in \mathbb N$ such that for any $r$-colouring of $[n] := \{1, \dots, n\}$ there is a ...
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1answer
125 views

Consider all colorings of the edges of K6 such that every edge is either colored red or blue…

Consider all colorings of the edges of K6 such that every edge is either colored red or blue. Prove or disprove: there always exist at least two monochromatic triangles in any 2-coloring of the edges ...
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1answer
43 views

Graph Ramsey Theory for Multiple Copies of Graphs

I had the following question from Graph Ramsey theory. Show that if $m \geq 2$, then $$ R((m+1)K _{3},K _{3})\geq R(mK _{3},K _{3}) + 3. $$ Thanks.
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3answers
197 views

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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2answers
81 views

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 ...
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0answers
31 views

Some small (and probably easy) implication from combinatorics paper

I'm reading https://people.math.osu.edu/bergelson.1/PolSz.pdf . Question is about part from (1.7) on page 16. Why $ \lambda(\chi_{1})=\lambda(\chi_{1}) $? There's no problem if in 4-th verse ...
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2answers
154 views

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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0answers
40 views

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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1answer
139 views

Six people at a party and its graph theory representation

I'm confused by this classic graph theory problem: Suppose there are six people at a party. Prove that it is always possible to find either three people who all know each other, or three people none ...
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1answer
64 views

$K_6$ contains at least two monochromatic $K_3$ graphs.

Let $K_n$ be a complete $n$ graph with a color set $c$ with $c=\{\text{Red}, \text{Blue}\}$. Every edge of the complete $n$ graph is colored either $\text{Red}$ or $\text{Blue}$. Since $R(3, 3)=6$, ...
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2answers
126 views

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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1answer
36 views

Ramsey Theory//Question of Notation

I RESOLVED THIS QUESTION FOR MYSELF, THANKS FOR VIEWING, SORRY We have this notation: $$ \mathcal A \to \mathcal B_k^{n} $$ Which means: \begin{align*} &(\forall A \in \mathcal A)(\forall ...
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1answer
66 views

A problem relying on van der Waerden's theorem, and the existence of sums divisible by a given number $n$

Say we are given a sequence of integers $\{a_i\}_{i \in \mathbb{N}}$, as well as a pair of integers $n, m$. How can we show that there always exist positive integers $s, r$ such that the sums $a_s + ...
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0answers
28 views

Practical use of Ramsey numbers [duplicate]

I am fascinated by the Ramsey number, but I was wondering, what are practical uses of the Ramsey number? Except for the party problem, I can not come up with something where it is useful for. Do you ...
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66 views

How prove this will be a length of $3$ arithmetic progression. [duplicate]

This problem is my frend give me: Given $122$ consecutive integers, show that randomly selected $33$ integer, of which there will be a length of $3$ arithmetic progression. hint: length of ...
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0answers
79 views

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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1answer
191 views

Ramsey Type problem (variant of people at a party)

There is $n$ people at a party. Prove that there are two people such that, of the remaining $n-2$ people, there are at least $\lfloor n/2\rfloor-1$ of them, each of whom either knows both or else ...
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1answer
62 views

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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0answers
42 views

2-colorings of arithmetic progression

Consider the following classical result: $\forall r \in \mathbb N : \exists N\in \mathbb N$ such that every 2-colored arithmetic progression of length N contains a monochromatic arithmetic ...
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3answers
229 views

Ramsey's theorem [closed]

I'm reading introduction to combinatorics and encountered an exercise I couldn't answer Let S be a set of six points in the plane, with no three of the points collinear. Color either red or blue each ...
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2answers
80 views

Reconciling two statements of Ramsey's Theorem.

In the appendix to Shelah's book on Classification Theory: THEOREM 2.1 (Ramsey's Theorem): (1) For any infinite ordered set $I$, and $n$-place function $f$ from $I$, with range of cardinality ...
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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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2answers
234 views

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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101 views

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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1answer
63 views

Prove a relation related to sets

In a city, among each pair of people, there can be exactly one of k different relationships (relationships are symmetric). A crowd is a set of three people in which every pair have the same relation. ...
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1answer
128 views

Accessible Applications of Graph Ramsey Theory

I am giving a short lecture series on graph Ramsey theory to a group of gifted high school seniors. The brief outline is to start with the "six people at a dinner party" question, transition into the ...
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0answers
125 views

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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1answer
43 views

n-coloring of a graph using Schur's Theorem

Using Schur Theorem I have to prove that for any $n\geq 1$ there is a number $f(n)$ such that if $A=${$2,3,4,...f(n)$} is partitioned into n sets $A_1$, $A_2$...$A_n$ there is a $j\in${1,2,...,n} ...
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1answer
101 views

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 $\mathbb N$.