Questions tagged [ramsey-theory]

Use for questions in Ramsey Theory, i.e. regarding how large a structure must be before it is guaranteed to have a certain property. Please be especially careful not to ask open questions in this tag.

Please be especially careful not to ask open questions in this tag.

Ramsey theory refers to questions of the form "how large must a structure be before it is guaranteed to have a certain property?" Often, the theme is that in a sufficiently large structure, a highly ordered substructure will appear.

A relatively simple example of a result in Ramsey theory is the Theorem on Friends and Strangers.

In any party of at least six people either at least three of them are (pairwise) mutual strangers or at least three of them are (pairwise) mutual acquaintances.

Other well-known results in Ramsey theory include:

  • Ramsey's theorem, which generalizes the Theorem on Friends and Strangers to larger subgroups than size $3$. Many other problems in Ramsey theory are variations on this result, and involve coloring graphs.
  • Schur's theorem, which says that for any $r$, there exists a sufficiently large $N$ such that whenever the integers $1, 2, \dots, N$ are each given one of $r$ colors, there will be three integers $x, y, x+y$ all of the same color. More generally, additive Ramsey theory deals with results about the integers and other additive groups, including results such as Van der Waerden's theorem.
  • The Hales–Jewett theorem which, informally, states that for any parameters $t$ and $r$ there is a sufficiently large dimension such that any $r$-coloring of a $t \times t \times \dots \times t$ grid contains a monochromatic line. More generally, Euclidean Ramsey theory deals with results about geometric objects.

Proofs in Ramsey theory often give extremely large bounds on how large a structure must be before it has the desired property.

A standard introduction to the area is the textbook Ramsey Theory by Graham, Rothschild, and Spencer.

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Lower bound for $R(3, 3,\ldots, 3)$

As part of learning Ramsey numbers I am trying to prove that $R(\underbrace{3, 3,\ldots, 3}_{k\text{ times}}) > 2^k$ using the constructive method. In order to do that one needs to colour the edges of a complete graph $K_{2^k}$ using k colours in…
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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 is $10$. Indeed, if we consider $K_9$ and color…
idm
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Party problem / Ramsey's theorem R(3,3)

I'm looking for an algorithm that solve Party problem. The party problem asks to find the minimum number of guests that must be invited so that at least 3 will know each other or at least 3 will not know each other. I know that the answer is 6 but i…
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Rectangular stained glass window with different colors

Suppose you have six squares of stained glass, all of different colors, and you would like to make a rectangular stained glass window in the shape of a 2 × 3 grid. How many different ways can you do this, taking symmetry into account? (Note that any…
Kyle
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Ramsey numbers: if $s_1 \leq s_2$ then $R(s_1,t)\leq R(s_2,t)$

I'm doing this little homework assignment on Ramsey numbers, the question is: Show that $$s_1 \leq s_2 \Rightarrow R(s_1,t)\leq R(s_2,t).$$ I've tried classifying it into these four cases: The graph on $R(s_1,t)$ vertices has a clique of size…
User32563
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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 is $4*(17-1)+2$. Would an explanation that doesn't…
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Combinatorics) Proof concerning Ramsey's Theory

Let $q_1$, $q_2$, ..., $q_k$, t be positive integers, where $q_1$≥t, $q_2$≥t, ..., $q_k$≥t. Let m be the largest of $q_1$, $q_2$, ..., $q_k$. Show that $r_t$(m, m, ..., m) ≥ $r_t$($q_1$, $q_2$, ..., $q_k$). Conclude that to prove Ramsey's theorem,…
ATP
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Ramsey Numbers and edge coloring

Show that for every $k \in\mathbb{N}$ there exists an $n \in\mathbb{N}$, where $n ≤ 3k!$ such that if $K_n$ is coloured in $k$ colours then we can find in $K_n$ a triangle whose edges are of the same colour. Thanks.
O L
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Modification of the Ramsey number

Let us denote by $n=r(k_1,k_2,\ldots,k_s)$ the minimal number of vertices such that for every coloring of the edges of the complete graph $K_n$ by $s$ different colors, there is some color $1\le i\le s$ such that the $i$-th graph contains a $k_i$…
Tai
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Monochromatic Solutions

I recently came across this paper: http://borisalexeev.com/pdf/foxgraham.pdf "On Minimal Colorings Without Monochromatic Solutions To a Linear Equation" Can someone explain in clearer terms what it means for a linear equation to not have a…
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Is there some non-classical logic where the van der Waerden theorem does not apply?

The van der Waerden theorem is a theorem in the branch of mathematics called Ramsey theory which states that for any given positive integers r and k, there is some number N such that if the integers {1, 2, ..., N} are colored, each with one of r…
vengaq
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Draw a regular graph on 8 vertices such that it does not have a K3 , and has no independent set of cardinality 4.

Draw a regular graph(one which has all vertices of equal degree) on 8 vertices such that it does not have a triangle and has no independent set of cardinality 4. I'm wondering if this is even possible? I've tried various graphs but this seems…
thedumbkid
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How to show that $g:2^M\to 2^\mathbb{N}$ defined by $g(A) = X\cup A$ is continuous?

In Galvin and Prikry's paper, they inroduce completely Ramsey sets. Definition $5$: A set $S\subseteq 2^\mathbb{N}$ is completely Ramsey if $f^{-1}(S)$ is Ramsey for every continuous mapping $f:2^\mathbb{N}\to 2^\mathbb{N}.$ Question: What is a…
Idonknow
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Van der Waerden's theorem in $\mathbb{Z}^2$

Let $\mathbb{Z}$ be the set of whole numbers and $l,m\in N$. Let's color all elements of $\mathbb{Z}\times\mathbb{Z}$ in $k$ different colors. Prove that we can find two aritmetic progressions $A$ and $B$ of length $l$ and $m$ in $\mathbb{Z}$ such…
Oldboy
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Ramsey numbers Olympiad combo

Show that if l,s are positive integers, then $r(l,s) \leq {l + s - 2 \choose l - 1} $
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