## Abstract

The Ramsey theorem says that for any countably infinite undirected clique whose edges are colored by a finite number of colors, there is an infinite subclique whose edges are colored by a single color. In this note, we generalize the theorem to a situation where the colors form a compact metric space.

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By [*A*]^{2} we denote the family of size two subsets of *A*, interpreted as edges in the undirected graph with nodes *A*. If *K* is a metric space and *A* is a set, we say that *k*∈*K* is the limit of a function *f*:[*A*]^{2}→*K* if for any *ϵ*>0, there is some finite subset *C*⊆*A* such that the image of *f*, when restricted to [*A*−*C*]^{2}, is contained in the ball of radius *ϵ* and center *k*.

*Let*
*K*
*be a metric compact space*, *B*
*a countably infinite set*, *and*
*f*:[*B*]^{2}→*K*. *Then there exists a*
*k*∈*K*
*and an infinite subset*
*A*⊆*B*
*such that the restriction of*
*f*
*to* [*A*]^{2}
*has limit*
*k*.

Ramsey’s theorem is a special case of the above theorem, when *K* is equipped with a discrete metric. The proof is a simple adaptation of the proof of Ramsey’s theorem.

Without loss of generality we assume that *B*=*ω*. By induction, we will construct a sequence of sequences *a*
^{0},*a*
^{1},*a*
^{2},…∈*B*
^{ω}, such that *a*
^{n} is a subsequence of *a*
^{n−1}. We write \(a^{n}_{m}\) for the *m*-the element of the *n*-th sequence *a*
^{n}.

Let \(a^{0}_{m} = m\). For *n*>0, let *a*
^{n} be a subsequence of *a*
^{n−1} such that \(a^{n}_{m} = a^{n-1}_{m}\) for *m*≤*n*, and the sequence

is convergent to some *k*
_{
n
}. This subsequence exists because *K* is compact. Moreover, we can choose the sequence *a*
^{n} so that for all *m*>*n* the distance from \(f(\{a^{n}_{n}, a^{n}_{m}\})\) to *k*
_{
n
} is at most 1/*n*.

Let (*q*
_{
n
}) be a sequence of numbers such that \(k_{q_{n}}\) is convergent to some *k* and moreover, \(\delta(k_{q_{n}}, k) \leq1/n\). Define *A* as \(\{a^{q_{n}}_{q_{n}} : n \in\omega\}\).

Let *n*<*m*. Since \(a^{q_{m}}\) is a subsequence of \(a^{q_{n}}\), we have that \(a^{q_{m}}_{q_{m}} = a^{q_{n}}_{z}\) for some *z*>*q*
_{
m
}>*q*
_{
n
}>*n*.

Since \(\delta(f(\{a^{q_{n}}_{q_{n}}, a^{q_{m}}_{q_{m}}\}), k) < 2/n\), the set *A* satisfies our claim. □

The theorem assumes that the space *K* is metric. We show a space *K* which is compact but not metric, and where the statement of the theorem fails. Let *K* be any compact space where not every sequence has a convergent subsequence. For instance, *K* can be an uncountable product of unit intervals. Take some sequence *a*
_{1},*a*
_{2},… in *K* which does not have any convergent subsequence, and define *f*:[*ω*]^{2}→*K* by *f*({*i*,*j*})=*a*
_{
i
}. The theorem, when applied to this function *f*, would imply that there is a converging subsequence of *a*
_{1},*a*
_{2},… .

By induction on *k*, one can generalize the theorem to functions *f* defined on *k*-element subsets.

In the following corollary, we see what happens when *K* has a semigroup structure. If *x*∈*K*
^{ω} is a sequence of elements from a semigroup *K*, we write *x*[*i*..*j*) for the multiplication of *x*
_{
i
}⋯*x*
_{
j−1}.

*If in Theorem *1, *the space*
*K*
*has a semigroup structure with continuous multiplication*, *B*=*ω*, *and*
*f*
*is obtained from a sequence*
*x*∈*K*
^{ω}
*by setting*
*f*({*i*,*j*})=*x*[*i*..*j*), *then the limit*
*k*
*is idempotent*, *i*.*e*. *k*=*k*⋅*k*.

Apply Theorem 1, yielding a set *A*={*u*
_{1}<*u*
_{2}<…}⊆*ω*. If the action is continuous, then *k* is idempotent, since

□

In some papers, e.g. [1], a *compact semigroup* is a semigroup *S* with compact topology such that the mapping *t*↦*s*⋅*t* is required to be continuous for each *s*∈*S*. This assumption is weaker than our assumption from Corollary 1 that the action in *S* is continuous.

However, the weaker assumption is not sufficient for idempotence of *s* in Corollary 1. Indeed, consider the semigroup *S*={0,1,2,…,*ω*,*ω*+1}, with the action *a*⊕*b*=min(*a*+*b*,*ω*+1), and the distance *δ*(*a*,*b*)=|*f*(*a*)−*f*(*b*)|, where *f*(*n*)=1/(*n*+1),*f*(*ω*)=0,*f*(*ω*+1)=−1. This semigroup is a compact semigroup in the meaning from [1]. Now, let *x*
_{
n
}=*n*. If we apply Corollary 1 to *x*, *s* has to be *ω*, but *ω* is not idempotent: *ω*⊕*ω*=*ω*+1.

## References

Furstenberg, H., Katznelson, Y.: Idempotents in compact semigroups and Ramsey theory. Isr. J. Math.

**68**(3), 257–270 (1989). doi:10.1007/BF02764984

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Communicated by Jean-Eric Pin.

M. Bojańczyk and S. Toruńczyk were supported by ERC Starting Grant “Sosna”. E. Kopczyński was supported by the Polish Ministry of Science Grant Nr. N N206 567840.

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Bojańczyk, M., Kopczyński, E. & Toruńczyk, S. Ramsey’s theorem for colors from a metric space.
*Semigroup Forum* **85**, 182–184 (2012). https://doi.org/10.1007/s00233-012-9404-4

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DOI: https://doi.org/10.1007/s00233-012-9404-4