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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15
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2answers
654 views

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 ...
3
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1answer
376 views

Ramsey Number Inequality: $R(\underbrace{3,3,…,3,3}_{k+1}) \le (k+1)(R(\underbrace{3,3,…3}_k)-1)+2$

I want to prove that: $$R(\underbrace{3,3,...,3,3}_{k+1}) \le (k+1)(R(\underbrace{3,3,...3}_k)-1)+2$$ where R is a Ramsey number. In the LHS, there are $k+1$ $3$'s, and in the RHS, there are $k$ $3'...
14
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1answer
1k 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 !
6
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1answer
363 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 ...
3
votes
1answer
186 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 $R(3, 3)=6$, ...
5
votes
2answers
215 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.
1
vote
1answer
316 views

Chromatic Triangles on a k17 graph

If the edges of the complete graph K17 (on 17 vertices with no three collinear) are each colored one of three colours can it be proven to have two or more monochromatic triangles?
5
votes
4answers
985 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 ...
3
votes
1answer
86 views

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 ...
1
vote
1answer
193 views

Best known bounds for Ramsey numbers

I realize a similar question has been asked before but what I want to know is a little different and is not answered by the link in the answer to that question. I am interested in knowing the best ...
3
votes
1answer
277 views

Ramsey Number proof: $R(3,3,3,3)\leq 4(R(3,3,3)-1) + 2$

I am trying to prove: $R(3,3,3,3)\leq 4(R(3,3,3)-1) + 2$ I am confused as to how one can go from a $4$ color problem to a $3$ color problem by multiplying and adding. edit: $R$ is the Ramsey ...
3
votes
1answer
127 views

Verify that R(p,2) = R(2,p) = p, where R is the Ramsey number

Verify that $R(p,2) = R(2,p) = p$, where $R$ is the Ramsey number It just seems obvious that $R(p,2) = R(2,p)$. But why do $R(p,2)$ and $R(2,p)$ both equal p?
3
votes
1answer
356 views

Amalgamation of graphs

I am trying to understand the definition of amalgamation of a $n+1$-partite graph as explained here(first few lines of page 4). We have a $n+1$-partite graph $G$ with partite sets $V_0,V_1,\cdots,V_n$ ...
3
votes
1answer
202 views

A “geometrical” representation for Ramsey's theorem

The [infinite] Ramsey theorem states that Let $n$ and $k$ be natural numbers. Every partition $\{X_1,\ldots ,X_k\}$ of $[\omega]^n$ into $k$ pieces has an infinite homogeneous set. Equivalently, ...
2
votes
1answer
106 views

Proving an 'obvious' Ramsey upperbound

Here I want to prove the following: $$R(s_{1},s_{2},\ldots ,s_{k}) < R(s_{1}+1,s_{2},\ldots ,s_{k})$$ For $s_{1},\ldots ,s_{k} \in \mathbb{N}$, $s_{i}\geq 2$. (Or can it hold with equality in some ...
1
vote
1answer
208 views

Girth and monochromatic copy of trees

Question: Prove that for every tree $T$ and every integer $g$ there exists a graph $G$ without cycles of length up to $g$ and such that every two-coloring of the edges of $G$ contains a monochromatic ...
5
votes
2answers
125 views

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 ...
5
votes
1answer
91 views

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 $n$ ...
9
votes
7answers
1k views

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 $A_1,\dots,A_k$ ...
6
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3answers
1k views

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 ...
4
votes
2answers
282 views

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 ...
4
votes
2answers
405 views

Given an infinite poset, show that it contains either a infinite chain or an infinite totally unordered set. [duplicate]

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, ...
7
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1answer
2k views

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 $v_1, v_2, \cdots , v_{13}$ ...
6
votes
2answers
171 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 $A$, $...
4
votes
1answer
141 views

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 \...
2
votes
2answers
176 views

How can we apply Ramsey's theorem to solve this problem on matrices?

This is from a problem set in a combinatorics course I am taking, and reads as follows: Let $m\geqslant 1$ be an integer. Prove there exists an $n\geqslant 1$ with he following property: every $M\...
2
votes
1answer
231 views

A sequence of $n^2$ real numbers which contains no monotonic subsequence of more than $n$ terms

I'm following a Combinatorics course at the moment, and have recent proved the Erdős–Szekeres Theorem (or, at least, some variation of): A sequence of length $n^2 + 1$ either contains an ...
2
votes
1answer
87 views

Coloring a Complete Graph in Three Colors, Proving that there is a Complete Subgraph

Color the edges of a complete graph on $n$ vertices $K_n$ in three colors (red,blue,yellow) such that at most $\dfrac{n^2}{k}$ are colored red ($k$ is some natural number). Prove that $K_n$ ...
6
votes
0answers
218 views

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 ...
4
votes
2answers
617 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 ...
2
votes
1answer
253 views

Upper bound for ramsey number $r(a_1,\ldots, a_m)$

I am looking for any (finite) upper bound of the ramsey number $r(a_1,\ldots, a_m)$. I can prove the well known fact for any positive integers $a,b$ there is a $c$ for which $c\ge r(a,b)$ by taking $c=...
2
votes
1answer
78 views

Ramsey lower bounds

I'm doing some self study on Ramsey graph theory. One of the first theorems concerning lower bounds shows that if $${n \choose k} \cdot \frac {1}{2^{( \frac {k}{2}-1)}} <1$$ then $N(k,k)> n$ ...
2
votes
3answers
94 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 $...
2
votes
2answers
148 views

Can the complete graph $K_9$, be 2-coloured with no blue $K_4$ or red triangles?

I am working on the following problem on 2-coloured complete graphs: $K_9$ is coloured red and blue and contains no red triangle and no blue $K_4$ then every vertex must have red degree 3 and ...
1
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0answers
98 views

Proving an inequality on Ramsey numbers by induction: $t_{r+1} \leq (r+1) (t_r - 1) + 2$ [duplicate]

Not sure how to proceed. I'm trying to prove that the following inequality is true. I know that $t_2 = 6$ and $t_3=17$ from the problem statement. The base case is obvious. $t_{r+1} \leq (r+1) (t_r -...
1
vote
1answer
263 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 ...
0
votes
1answer
66 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 ...