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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Closest pair of vectors in $\{0,1\}^n$

Suppose we are given $k$ points in $\{0,1\}^n$ (using Hamming distance as metric). Consider the two points that have the smallest distance between them. Does there exist any results bounding this distance? In particular I am interested in an…
Magnus Find
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Finding where in Ramsey's theorem one uses the Axiom of choice

Ramsey's Theorem for infinite graphs requires some choice but when looking at the proof it is not evident how choice is exactly used. Sketch of the proof: Given $c:[\omega]^2\rightarrow 2$ a coloring we construct a homogeneous set as follows.…
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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 potential couples there are; mark this number by…
Erel Segal-Halevi
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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 proof that $R(m,n)$ exists in general, and…
Austin Mohr
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Finding a set of $n-1$ languages, such that everyone speaks at least one language in the set

For any integer $n$, the following fact can be proven to be true: Given $n^n+1$ people, where each person speaks a distinct set of $n$ languages, such that any two of these people speak at least one language in common, there exists a set $T$ of…
Mike Earnest
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Binomial upper bound for the bi-color Ramsey numbers (Erdős-Szekeres)

The question: How did Erdös - Szekeres came up with a close form with a binomial for the upper bound: Where does the idea behind $R(2,2)=\binom{2+2-2}{2-1}$ - I do see that $R(2,2)=2$ - or $\binom{s+t-3}{s-1}\left(\text{or…
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Ramsey property and linear orders on $\kappa$

I have been trying to solve to prove the following statement: Let $\kappa$ be an uncountable cardinal. The following are equivalent: Every linear order of cardinality $\kappa$ has a suborder of order-type $\kappa$ or $\kappa^*$ ($\kappa$…
Darío G
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Does a red/blue coloring of the infinite subsets of $\mathbb{N}$ necessarily give an infinite monochromatic $M\subset \mathbb{N}$?

The infinite Ramsey theorem states that for any $n$, if all the subsets of $\mathbb{N}$ of size $n$ are colored red/blue, then there is an infinite $M$ all of whose subsets of size $n$ are monochromatic. My question is whether there is an analogue…
Spook
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If in all subsets of size k there exist at least one pair of elements in a relation...

I’ve recently got stuck on a theorem that seems to be true, but I am not sure if I am able to prove it: $\large\forall_{x\in\mathbb{P}_k(\mathbb{N})}\exists_{\left\{ y_1,y_2\right\}\subset x \land y_1 \neq y_2}\; \varphi(y_1,y_2) \rightarrow…
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A Combinatorics Problem with x+y = z

The numbers $\{1,2,...,2005\}$ are divided into $6$ disjoint subsets. Prove that for one of them we can find $x,y,z$ elements in it, not necessarily distinct such that $x + y = z$. I have no idea how to solve this or where to start. Also I can't…
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in every coloring $1,...,n$ there are distinct integers $a,b,c,d$ such that $a+b+c=d$

Prove that for every $k$ there is a finite integer $n = n(k)$ so that for any coloring of the integers $1, 2, . . . , n$ by $k$ colors there are distinct integers $a, b, c$ and $d$ of the same color satisfying $a + b + c = d$
AvR
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Ramsey theory, finite colourings of $\mathbb{N}$ and infinite monochromatic sets

I am trying to show that the following statement is false: whenever $\mathbb{N}$ is finitely coloured by $c: \mathbb{N} \to \{1,\ldots,k\}$ (in the sense of Ramsey theory), there exists an infinite set $x_1
Spyam
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Prove that $x\ge n(n-1)(n-3)/8$, where $x$ is the number of $4$-cycles in a graph on $n$ vertices with at least $\frac12\binom{n}{2}$ edges.

Prove that $x\ge \dfrac{n(n-1)(n-3)}8$, where $x$ is the number of $4$-cycles in a graph on $n$ vertices with at least $\displaystyle\frac12\,\binom{n}{2}$ edges. If there are no $4$-cycles in a graph the edges in $G$, then the number of edges…
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Is there a simpler proof of Van der Waerden's Theorem when there are only two colors?

http://en.wikipedia.org/wiki/Van_der_Waerden%27s_theorem The usual approach is to induct on the length of the arithmetic progression, which is difficult to simplify directly to the case of two colors. Does anyone know a different approach? Thank…
Ben Derrett
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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 too, but none longer than four. How long does a…
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