Structural Ramsey theory explained
In mathematics, structural Ramsey theory is a categorical generalisation of Ramsey theory, rooted in the idea that many important results of Ramsey theory have "similar" logical structures. The key observation is noting that these Ramsey-type theorems can be expressed as the assertion that a certain category (or class of finite structures) has the Ramsey property (defined below).
Structural Ramsey theory began in the 1970s[1] with the work of Nešetřil and Rödl, and is intimately connected to Fraïssé theory. It received some renewed interest in the mid-2000s due to the discovery of the Kechris–Pestov–Todorčević correspondence, which connected structural Ramsey theory to topological dynamics.
History
is given credit for inventing the idea of a Ramsey property in the early 70s. The first publication of this idea appears to be Graham, Leeb and Rothschild's 1972 paper on the subject.[2] Key development of these ideas was done by Nešetřil and Rödl in their series of 1977[3] and 1983[4] papers, including the famous Nešetřil–Rödl theorem. This result was reproved independently by Abramson and Harrington,[5] and further generalised by .[6] More recently, Mašulović[7] [8] [9] and Solecki[10] [11] [12] have done some pioneering work in the field.
Motivation
This article will use the set theory convention that each natural number
can be considered as the set of all natural numbers less than it: i.e.
. For any set
, an
-colouring of
is an assignment of one of
labels to each element of
. This can be represented as a function
mapping each element to its label in
(which this article will use), or equivalently as a partition of
into
pieces.
Here are some of the classic results of Ramsey theory:
, there exists
such that for every
-colouring
of all the
-element subsets of
, there exists a subset
, with
, such that
is
-monochromatic.
, there exists
such that for every
-colouring
of
, there exists a
-monochromatic arithmetic progression
\{a,a+d,a+2d,\ldots,a+(m-1)d\}\subseteqn
of length
.
. A
-parameter word of length
over
is an element
w\in(L\cup\{x0,x1,\ldots,xk-1\})n
, such that all of the
appear, and their first appearances are in increasing order. The set of all
-parameter words of length
over
is denoted by
. Given
and
, we form their
composition stylew\circv\in[L]\binom{n}{k}
by replacing every occurrence of
in
with the
th entry of
.
Then, the Graham–Rothschild theorem states that for every
, there exists
such that for every
-colouring
style\Delta:[L]\binom{n}{k}\tor
of all the
-parameter words of length
, there exists
, such that
stylew\circ[L]\binom{m}{k}=\{w\circv:v\in[L]\binom{m}{k}\}
(i.e. all the
-parameter subwords of
) is
-monochromatic.
, there exists
such that for every
-colouring
of
, there exists a subset
, with
, such that
, and
style\operatorname{FS}(A)=\{\sumkk:B\inl{P}(A)\setminus\varnothing\}
is
-monochromatic.
These "Ramsey-type" theorems all have a similar idea: we fix two integers
and
, and a set of colours
. Then, we want to show there is some
large enough, such that for every
-colouring of the "substructures" of size
inside
, we can find a suitable "structure"
inside
, of size
, such that all the "substructures"
of
with size
have the same colour.
What types of structures are allowed depends on the theorem in question, and this turns out to be virtually the only difference between them. This idea of a "Ramsey-type theorem" leads itself to the more precise notion of the Ramsey property (below).
The Ramsey property
Let
be a category.
has the
Ramsey property if for every natural number
, and all objects
in
, there exists another object
in
, such that for every
-colouring
\Delta:\operatorname{Hom}(A,D)\tor
, there exists a
morphism
which is
-monochromatic, i.e. the set
f\circ\operatorname{Hom}(A,B)=\{f\circg:g\in\operatorname{Hom}(A,B)\}
is
-monochromatic.
[9] Often,
is taken to be a class of finite
-structures over some fixed
language
, with
embeddings as morphisms. In this case, instead of colouring morphisms, one can think of colouring "copies" of
in
, and then finding a copy of
in
, such that all copies of
in this copy of
are monochromatic. This may lend itself more intuitively to the earlier idea of a "Ramsey-type theorem".
There is also a notion of a dual Ramsey property;
has the dual Ramsey property if its
dual category
has the Ramsey property as above. More concretely,
has the
dual Ramsey property if for every natural number
, and all objects
in
, there exists another object
in
, such that for every
-colouring
\Delta:\operatorname{Hom}(D,A)\tor
, there exists a
morphism
for which
\operatorname{Hom}(B,A)\circf
is
-monochromatic.
Examples
- Ramsey's theorem: the class of all finite chains, with order-preserving maps as morphisms, has the Ramsey property.
for
,
, the Ramsey property holds for
.
let
be a finite alphabet, and for each
, let
be a set of
variables. Let
be the category whose objects are
for each
, and whose morphisms
, for
, are functions
which are rigid and surjective on Xk\subseteqAk=\operatorname{codom}f
. Then,
has the dual Ramsey property for
(and
, depending on the formulation).- Graham–Rothschild theorem: the category
defined above has the dual Ramsey property.
The Kechris–Pestov–Todorčević correspondence
In 2005, Kechris, Pestov and Todorčević[13] discovered the following correspondence (hereafter called the KPT correspondence) between structural Ramsey theory, Fraïssé theory, and ideas from topological dynamics.
Let
be a
topological group. For a topological space
, a
-flow (denoted
) is a
continuous action of
on
. We say that
is
extremely amenable if any
-flow
on a compact space
admits a
fixed point
, i.e. the
stabiliser of
is
itself.
, its
automorphism group
can be considered a topological group, given the topology of
pointwise convergence, or equivalently, the
subspace topology induced on
by the space
with the
product topology. The following theorem illustrates the KPT correspondence:
Theorem (KPT). For a Fraïssé structure
, the following are equivalent:- The group
of automorphisms of
is extremely amenable.- The class
has the Ramsey property.
See also
Notes and References
- Van Thé. Lionel Nguyen. 2014-12-10. A survey on structural Ramsey theory and topological dynamics with the Kechris–Pestov–Todorcevic correspondence in mind. 1412.3254. math.CO.
- Graham . R. L.. Leeb . K.. Rothschild . B. L.. 1972. Ramsey's theorem for a class of categories. Advances in Mathematics. 8. 3. 417–433. 10.1016/0001-8708(72)90005-9. free. 0001-8708.
- Nešetřil. Jaroslav. Rödl. Vojtěch. May 1977. Partitions of finite relational and set systems. Journal of Combinatorial Theory, Series A. 22. 3. 289–312. 10.1016/0097-3165(77)90004-8. 0097-3165. free.
- Nešetřil. Jaroslav. Rödl. Vojtěch. 1983-03-01. Ramsey classes of set systems. Journal of Combinatorial Theory, Series A. 34. 2. 183–201. 10.1016/0097-3165(83)90055-9. 0097-3165. free.
- Abramson. Fred G.. Harrington. Leo A.. September 1978. Models Without Indiscernibles. The Journal of Symbolic Logic. 43. 3. 572. 10.2307/2273534. 0022-4812. 2273534. 1101279.
- Prömel. Hans Jürgen. July 1985. Induced partition properties of combinatorial cubes. Journal of Combinatorial Theory, Series A. 39. 2. 177–208. 10.1016/0097-3165(85)90036-6. 0097-3165. free.
- Masulovic . Dragan. Scow . Lynn. 2017. Categorical equivalence and the Ramsey property for finite powers of a primal algebra. Algebra Universalis. 78. 2. 159–179. 10.1007/s00012-017-0453-0. 1506.01221. 125159388.
- Masulovic . Dragan. 2018. Pre-adjunctions and the Ramsey property. European Journal of Combinatorics. 70. 268–283. 10.1016/j.ejc.2018.01.006. 1609.06832. 19216185.
- Mašulović . Dragan. 2020. On Dual Ramsey Theorems for Relational Structures. Czechoslovak Mathematical Journal. 70. 2. 553–585. 10.21136/CMJ.2020.0408-18. 1707.09544. 125310940.
- Solecki. Sławomir. August 2010. A Ramsey theorem for structures with both relations and functions. Journal of Combinatorial Theory, Series A. 117. 6. 704–714. 10.1016/j.jcta.2009.12.004. 0097-3165. free.
- Solecki. Slawomir. 2011-04-20. Abstract approach to finite Ramsey theory and a self-dual Ramsey theorem. 1104.3950. math.CO.
- Solecki. Sławomir. 2015-02-16. Dual Ramsey theorem for trees. 1502.04442. math.CO.
- Kechris. A. S.. Pestov. V. G.. Todorcevic. S.. February 2005. Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups. Geometric and Functional Analysis. 15. 1. 106–189. 10.1007/s00039-005-0503-1. 6937893. 1016-443X.