# Tagged Questions

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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### 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 $K_n$ ...
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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 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 $N(k,k)> n$ ...
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### Ramsey Numbers and the Probabilistic Method

I am trying to prove that if there exists a real number $0 < p <1$ such that $$\binom{n}{k}p^k + \binom{n}{q}(1-p)^q < 1$$ for some $k, q \in [n],$ Then $R(k,q) > n.$ I see by the ...
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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 like....
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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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### 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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### 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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### 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 ...
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.