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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What is the proof of the "thin set theorem"? A result in infinite Ramsey theory.

OK so here's a precise question: Is it true that for every integer $k\geq1$ and every $f:\mathbb{Z}^k\to\mathbb{Z}$, there is some infinite subset $A\subseteq\mathbb{Z}$ such that $f(A^k)$ is not all of $\mathbb{Z}$? Here $A^k$ is the obvious subset…
Kevin Buzzard
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intermediate step in proving old Ramsey lower bound

Let $r(n,n)=r(n)$ be the usual Ramsey number of a graph. It is known that $$\frac{1}{e\sqrt{2}}n2^{n/2}
user24503
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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 $A$, $B$, and $A \cup B$ all have the same color.
Hai Phan
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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 $v_1, v_2, \cdots , v_{13}$ with edges coloured with either red or blue. By…
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Understanding various definitions of TREE($n$) in Friedman's finite form of Kruskal's tree theorem.

I was reading the Wikipedia article on Friedman's finite form of Kruskal's tree theorem, and am interested in the large numbers TREE(n). I would like to verify TREE(2)=3 myself, but find conflicting definitions of the conditions on the…
Mark S.
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$K_6$ 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 $R(3, 3)=6$, the $K_6$ graph must contain at least one…
www.data-blogger.com
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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
ziang chen
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Syndeticity- and thickness-preserving bijections of $\mathbb N$

Let me recall some definitions: a set $A \subseteq \mathbb N$ is: syndetic if it intersects every large enough interval, i.e. if $\exists \ell \in \mathbb N^* : \forall k \in \mathbb N, A \cap ⟦ k, k+\ell - 1 ⟧ \neq\varnothing$ ; thick if it…
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Euclidian plane $\pi$ with all points either red, green or blue

In the Euclidian plane $\pi$ all points are either red, green or blue. Prove that you can select three points $A$, $B$ and $C$ from plane $\pi$ so that the the triangle $ABC$ satisfies all the following conditions: Points $A,B,C$ have the same…
Oldboy
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An upper bound for poset Ramsey number $2n \le R(Q_n,Q_n) \le n^2+2n$

For each integer $n,m \ge 1$ $2n \le R(Q_n,Q_n) \le n^2+2n$. Here $Q_n$ denotes a Boolean lattice of dimension $n$ and $R(P,P')$ or $R_{\dim_2}(P,P')$ is the smallest $N$ such that any red/blue coloring of $Q_N$ contains either red copy or $P$ or a…
Verse
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Erdős Probabilistic method

My question is based on the Erdos probabilistic method. I am trying to read from the paper here. The proof of Theorem 1 contains the statement Since a block sequence is monochromatic with probability $2^{1−k}$, it follows from the linearity of…
Shahab
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What is the explanation for the $64$ in Graham's number $g_{64}$?

As in, why does the iteration of the function until $g_{64}$ guarantee this property that defines Graham's number? Why was this number chosen? If I had to guess (emphasis on guess), I'd say that the Ramsey theoretical problem involving Graham's…
MCT
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An upper bound for a graph Ramsey number

I am trying to prove the following result, given as an exercise in my book: $r(K_m+\bar{K_n},K_p+\bar{K_q})\le\binom{m+p-1}{m}n+\binom{m+p-1}{p}q$. Here $r(G,H)$ denotes the Ramsey number for the graphs $G$ and $H$, i.e. the smallest positive…
Shahab
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Prove that $r(k,k) + k \leq r(k + 1, k + 1) $

Prove that $r(k,k) + k \leq r(k + 1, k + 1)$, where $r(k,l)$ is the minimum number of vertexes in a Graph, where we have a clique with $k$ vertexes or a stable set with $l$ vertexes. There are three theorems I know in the area. I tried to prove it…
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What's the difference between Ramsey theory and Extremal graph theory?

Wikipedia teaches us that problems in Ramsey theory typically ask a question of the form: "how many elements of some structure must there be to guarantee that a particular property will hold?" It also teaches us that "Extremal graph theory studies…
Gadi A
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