Comptes Rendus Mathématique

Thomas Leistner

Thomas Munn

Comptes Rendus
Mathématique
Thomas Leistner and Thomas Munn
Completeness of certain compact Lorentzian locally symmetric spaces
Volume 361 (2023), p. 819-824
Published online: 11 May 2023
https://doi.org/10.5802/crmath.449
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 819-824
https://doi.org/10.5802/crmath.449
Geometry and Topology / Géométrie et Topologie
Completeness of certain compact Lorentzian
locally symmetric spaces
Thomas Leistner∗, a and ThomasMunnb
a School of Mathematical Sciences, University of Adelaide, SA 5005, Australia
b Lund University, Faculty of Science, Centre for Mathematical Sciences, Box 118,
22100 Lund, Sweden
E-mails: thomas.leistner@adelaide.edu.au, thomas.munn@math.lu.se
Abstract. We show that a compact Lorentzian locally symmetric space is geodesically complete if the
Lorentzian factor in the local de Rham–Wu decomposition is of Cahen–Wallach type or if the maximal flat
factor is one-dimensional and time-like. Our proof uses a recent result by Mehidi and Zeghib and an earlier
result by Romero and Sánchez.
Keywords. Lorentzian manifolds, Lorentzian symmetric spaces, geodesic completeness.
Mathematical subject classification (2010). 53C50, 53C35.
Funding. This project has received funding from the Australian Research Council (DP190102360).
Manuscript received 15 November 2022, revised 22 November 2022, accepted 23 November 2022.
1. Introduction and statement of result
In contrast to Riemannian manifolds, a compact Lorentzian manifold is not necessarily geodesi-
cally complete — the fundamental example is the incomplete Clifton–Pohl torus, see [12, Exam-
ple 7.16]. In fact, the set of incomplete Lorentzian metrics on the 2-torus is dense in the set of
Lorentizan metrics [14]. Hence, the question whether or not a compact Lorentzian manifolds is
complete is fundamental in Lorentzian geometry, especially when addressing classification prob-
lems for compact manifolds.
One condition that ensures that a compact Lorentzian manifold (M , g ) is complete is a
sufficient amount of global symmetry: Marsden showed that any compact homogeneous semi-
Riemannian manifold is complete [9]. This includes symmetric spaces, i.e. manifolds for which
each point p ∈ M is a fixed point of an isometry whose differential at p is − Id. In a striking
generalisation of Marsden’s result in the Lorentzian context, Romero and Sánchez showed that
(M , g ) is complete if it admits a time-like conformal Killing vector field [15], so a rather small
amount of global symmetry. In the case of only local homogeneity, the question is much harder
to answer. Only when the dimension of M is 3, Dumitrescu and Zeghib [5] were able to show
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
820 Thomas Leistner and Thomas Munn
that if (M , g ) is compact and locally homogeneous, then (M , g ) is isometric to a quotient of a
Lorentizan homogeneous space G/H by a discrete subgroup Γ of G acting properly and freely on
G/H , and consequently is complete.
Other authors have imposed conditions on the curvature of compact Lorentzian manifold
(M , g ): if (M , g ) is flat [4] or, more generally, has constant sectional curvature [7], then it is
complete. In a combination of global and curvature assumptions, in [8] it was shown that (M , g )
is complete if it is a pp-wave, i.e. if (M , g ) admits a parallel null vector field and satisfies a
certain curvature condition. Very recently this result was generalised by Mehidi and Zeghib [10],
who showed that the curvature condition can be dropped, i.e. that the existence of a parallel
null vector field on a compact Lorentzian manifold implies that the manifold is complete.
These results show that compactness and certain types of Lorentzian special holonomy imply
completeness. Another result that links completeness to the holonomy group was given in [1].
Of course, assumptions on the homogeneity and on the curvature are not independent of each
other. Examples are locally symmetric spaces, which are defined by the condition that the curva-
ture tensor is parallel. A locally symmetric space (M , g ) is locally isometric to a specific symmet-
ric space, the model space for (M , g ). Since symmetric spaces are complete, by the de Rham–Wu
decomposition theorem [13, 16], their universal cover decomposes into indecomposable sym-
metric spaces, i.e. those that cannot be further decomposed into a semi-Riemannian product.
Hence, every locally symmetric space has a local product decomposition into locally symmet-
ric spaces that are locally isometric to an indecomposable symmetric space. We refer to this as
the local de Rham–Wu decomposition of a locally symmetric space, see Section 2 for more details.
It has been shown by Cahen and Wallach [3] that there are only two types of indecomposable
Lorentzian symmetric spaces: the spaces of nonzero constant sectional curvature — de Sitter
and anti-de Sitter space — and the Cahen–Wallach spaces, see Section 2 for their definition.
If one can show that all compact locally symmetric spaces are complete, then they are isomet-
ric quotients of their model spaces and their classification can be reduced to the classification of
compact Clifford–Klein forms, and hence to the algebraic problem of classifying co-compact iso-
metric group actions (see for example the results on compact quotients of Cahen–Wallach spaces
and further references on compact Lorentzian Clifford–Klein forms in [6]). However, so far this
has only been achieved in dimension 3 (in [5] as a special case of locally homogeneous spaces)
and for the indecomposable spaces: the results in [4], [7] and [8] imply that a compact, indecom-
posable Lorentzian locally symmetric space is geodesically complete. In the second author’s the-
sis [11] this was generalised to locally symmetric spaces which are local products of Euclidean
space and a Cahen–Wallach space. Using the same method as in [11], the new result in [10], as
well as the result in [15], here we generalise this further.
Theorem 1. Let (M , g ) a compact Lorentzian locally symmetric space whose local de Rham–Wu
decomposition satisfies one of the following conditions:
(i) the Lorentzian factor is of Cahen–Wallach type, or
(ii) the maximal flat factor is one-dimensional and time-like.
Then the time-orientable cover of (M , g ) admits a parallel vector field and hence (M , g ) is geodesi-
cally complete.
This leaves open the cases when (M , g ) does not have constant sectional curvature but the
Lorentzian factor in the local de Rham–Wu decomposition of (M , g ) does. The simplest examples
of this situation are those when (M , g ) is locally isometric to a product of Minkowski space and
a sphere, or to a product of (anti-)de Sitter space with Euclidean space or a sphere. In [11] an
attempt was made to apply the approach in [4] and [7] to such cases, but the methods are too
specific to the constant curvature model spaces and do not easily transfer to products thereof.
Thomas Leistner and Thomas Munn 821
2. The local de Rham–Wu decomposition of a Lorentzian locally symmetric space
In this section we will clarify what we mean by the local de Rham–Wu decomposition of a locally
symmetric space and we will recall the notion of a Cahen–Wallach space.
We call a symmetric space (M , g ) indecomposable if the connected component of its holo-
nomy group does not admit a non-trivial invariant subspace of the tangent space on which the
metric remains non-degenerate. Since symmetric spaces are complete, by the de Rham–Wu de-
composition theorem [13, 16], this is equivalent to the condition that the universal cover (M̃ , g̃ )
of (M , g ) does not decompose into a semi-Riemannian product. Hence, the universal cover of a
symmetric space is globally isometric to a semi-Riemannian product of simply connected inde-
composable symmetric spaces. Note that a Riemannian symmetric space (M , g ) is indecompos-
able if and only if the connected component of its holonomy group acts irreducibly, i.e. does not
admit any non-trivial invariant subspace of the tangent space.
Cahen and Wallach [3] have shown that an indecomposable Lorentzian symmetric space (M , g )
either has nonzero constant sectional curvature or is universally covered by M̃ =Rn+2 with
g̃ = 2dtdv +xaQab xbdt 2 +δabdxadxb , (1)
where Qab is a symmetric n ×n-matrix with nonzero determinant and (t , v, x1, . . . , xn) are global
coordinates on M̃ = Rn+2. Then (M̃ , g̃ ) is called a Cahen–Wallach space. A Cahen–Wallach space
admits a parallel null vector field ∂∂v , and the only non-vanishing terms (up to symmetry) of its
curvature tensor are
R
(
∂
∂t
,
∂
∂xa
,
∂
∂t
,
∂
∂xb
)
=Qab , (2)
for a,b = 1, . . . ,n. The Ricci tensor ρ, as an endomorphism of T M̃ , is
ρ = q ∂
∂v
⊗dt , (3)
where q is the trace of the matrix Qi j . In particular, ρ2 = 0. Two Cahen–Wallach spaces are
isometric if their Qi j ’s have the same spectrum with the same multiplicities.
Applying the de Rham–Wu decomposition to the universal cover of a Lorentzian symmetric
space (M , g ), we obtain that (M̃ , g̃ ) is globally isometric to a product of
(a) an indecomposable Lorentzian symmetric space, i.e. of nonzero constant sectional cur-
vature or a Cahen–Wallach space, or (R,−dt 2),
(b) irreducible Riemannian symmetric spaces, and
(c) Euclidean space.
Of course, (b) or (c) may not be present in the decomposition. Moreover, any such a decompo-
sition has a maximal flat factor, which is either empty or equal to (R,−dt 2), Minkowski space, or
Euclidean space.
A locally symmetric space (M , g ) is a semi-Riemannian manifold with parallel curvature
tensor, ∇R = 0. Every locally symmetric space is locally isometric to a globally symmetric space,
and hence locally isometric to a product of indecomposable symmetric spaces. We refer to this
local product decomposition as the local de Rham–Wu decomposition of (M , g ). We call a locally
symmetric space indecomposable if it is locally isometric to an indecomposable symmetric
space. Consequently, an indecomposable Lorentzian locally symmetric space either has constant
sectional curvature or is locally isometric to a Cahen–Wallach space. The latter means that there
are local coordinates such that the metric is of the form (1). Indecomposable compact Lorentzian
locally symmetric spaces have been shown to be complete, the ones with constant sectional
curvature in [4, 7] and the ones of Cahen–Wallach type in [8].
Since a locally symmetric Lorentzian symmetric space (M , g ) is locally isometric to a product
of of symmetric spaces as in (a), (b), and (c), the tangent space at a point decomposes into
822 Thomas Leistner and Thomas Munn
orthogonal subspaces that correspond to the tangent spaces of the local factors. Moreover, the
curvature tensor and the Ricci tensor of (M , g ) decompose accordingly into the sum of the
curvature tensors and Ricci tensors of the factors in the local decomposition. Since irreducible
Riemannian symmetric spaces of dimension > 1 are Einstein with nonzero Einstein constant,
this implies the following: if the local de Rham–Wu decomposition of (M , g ) does not contain
a Cahen–Wallach factor, then the Ricci tensor of (M , g ) is given by the sum of multiples of the
identity, and if it does contain a Cahen–Wallach factor, then the Ricci tensor is a sum of ρ as in (3)
and multiples of the identity.
3. Proof of Theorem 1
Let (M , g ) be a locally symmetric Lorentzian space of dimension m and let Holp be its holonomy
group at a point p ∈ M . Let R be the curvature tensor of (M , g ) and ρ the Ricci tensor considered
as an endomorphism of the tangent bundle. Since ∇R = 0, ∇ρ = 0, so both R and ρ are invariant
under Holp . We define two vector distributions, the nullity of R,
N := {X ∈ T M | X R = 0},
and the generalised kernel of ρ,
K := {X ∈ T M ∣∣ ∃ m ∈N : ρm(X ) = 0}.
Clearly, both N and K are invariant under Holp , since for H ∈ Holp we have
R(HN |p , X ,Y , Z ) = R(N |p , H−1X , H−1Y , H−1Z ) = 0
and
ρ(H X ) = H−1ρ(X ),
for all X ,Y , Z ∈ Tp M . This implies that also the orthogonal distributions N ⊥ and K ⊥ are
invariant under Holp . We clearly have that N ⊂K .
The remarks at the end of the previous section imply that in the case when the local de Rham–
Wu decomposition of (M , g ) does not contain a Cahen–Wallach factor, we have that N = K ,
T M =N ⊕N ⊥, and N ̸= 0 only if the local de Rham–Wu decomposition contains a flat factor. If
the flat factor is one-dimensional and time-like, we have that N is a time-like line bundle and an
arbitrary element H ∈ Holp in the decomposition Tp M =N |p ⊕N ⊥|p is of the form
H =
(±1 0
0 A
)
,
with A ∈ O(m −1). Hence, the time-orientable cover of (M , g ) admits a time-like parallel vector
field and is compact if M is compact. A parallel vector field is of course conformally Killing, so we
can apply the result in [15] to conclude that (M , g ) is complete. This proves case (ii) in Theorem 1.
From now on assume that the local de Rham–Wu decomposition of (M , g ) does contain a
Cahen–Wallach factor. Since ∂∂v is a local parallel null vector field, and with (2), we get that
L := N ∩N ⊥ ⊂ T M
is not trivial, and hence a bundle of null lines that is invariant under the holonomy group of
(M , g ). In particular, we have the following holonomy invariant filtrations
L ⊂ N ⊂ K , L ⊂ N ⊥∩K ⊂ K .
Moreover, since the Ricci-tensor of a Cahen–Wallach space is nilpotent and the irreducible
Riemannian factors are Einstein with non-zero Einstein constant, K ⊥ is non-degenerate, so that
T M =K ⊕K ⊥
Thomas Leistner and Thomas Munn 823
is a holonomy invariant decomposition of T M . At p ∈ M , we have the Holp -invariant decompo-
sition
Tp M =K |p ⊕K ⊥|p ,
with K |p = T ⊕T , where T ̸= {0} is the tangent space at p of the local Cahen–Wallach factor and
T the (possibly trivial) tangent space of the Euclidean (i.e. flat and Riemannian) factor. Note that
the decomposition K |p = T ⊕T may not be Holp -invariant. In order to describe the action of
Holp on Tp M , we denote by (t , v, x1, . . . xn) the coordinates on the local Cahen–Wallach factor as
in (1) and fix the following basis of K |p ,
e− = ∂
∂v
, e1, . . . , en , en+1, . . . , en+k , e+ =
∂
∂v
−xaQab xb
∂
∂t
,
where ea = ∂∂xa for a = 1, . . . ,n and en+b for b = 1, . . . ,k an orthonormal basis for the Euclidean
metric on T . In such a basis, the metric restricted to K |p is of the form
g |p =
0 0 10 1 0
1 0 0
 .
In particular we have
T = span{e−,e1 . . . ,en ,e+}, T = span{en+1 . . . ,en+k },
and that
L |p =R ·e−, (N ⊥∩K )|p = span{e−,e1, . . . ,en}, N |p = span{e−,en+1, . . . ,en+k },
are Holp -invariant. Then, in the Holp -invariant decomposition Tp M =K |p ⊕K ⊥|p , an arbitrary
element H ∈ Holp is of the form
H =
(
F 0
0 G
)
,
where F , in the above basis, writes as
F =

λ ∗ ∗ ∗
0 A 0 u
0 0 B v
0 0 0 λ−1
 ,
with A ∈ O(n), B ∈ O(k), u ∈ Rn , v ∈ Rk , 0 ̸= λ ∈ R, and the ∗-terms are determined by the other
terms. In particular, we have that
H(ea) = F (ea) = Aa beb mod e−, H(e+) = F (e+) =λe++uaea + vc ec mod e−,
where the unbarred indices a,b,c . . . run from 1 to n and the barred indices a,b,c, . . . from n +1
to n +k. Then, with (2) and since e− R = 0 and ec R = 0, we have
(H−1 ·R)(ea ,e+,eb ,e+) = R(Hea , He+, Heb , He+) = R(Fea ,Fe+,Feb ,Fe+) =
Aa cQcd Ab
d
λ2
.
Hence, the holonomy invariance of R,
(H−1 ·R)(ea ,e+,eb ,e+) = R(ea ,e+,eb ,e+) =Qab ,
yields the matrix equation for Q = (Qab),
λ2Q = A⊤Q A.
Since A ∈ O(n) and det(Q) ̸= 0, this implies that λ = ±1. This however shows that the time-
orientable cover of (M , g ) admits a global parallel null vector field, see [2, Proposition 2] for
details.
824 Thomas Leistner and Thomas Munn
If we now assume that M is compact, the time-orientable cover is compact and admits a global
parallel null vector field, and hence, by the result in [10] is geodesically complete. This implies
that (M , g ) is geodesically complete. □
Acknowledgements
A special case of Theorem 1 was obtained in the second author’s thesis [11], written under
supervision of the first author. We would like to thank Michael Eastwood for taking the role of
co-supervisor and for helpful discussions and comments.
References
[1] L. A. Aké Hau, M. Sánchez, “Compact affine manifolds with precompact holonomy are geodesically complete”, J.
Math. Anal. Appl. 436 (2016), no. 2, p. 1369-1371.
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p. 797-828.
[3] M. Cahen, N. Wallach, “Lorentzian symmetric spaces”, Bull. Am. Math. Soc. 76 (1970), p. 585-591.
[4] Y. Carrière, “Autour de la conjecture de L. Markus sur les variétés affines”, Invent. Math. 95 (1989), no. 3, p. 615-628.
[5] S. Dumitrescu, A. Zeghib, “Géométries lorentziennes de dimension 3: classification et complétude”, Geom. Dedicata
149 (2010), p. 243-273.
[6] I. Kath, M. Olbrich, Compact quotients of Cahen-Wallach spaces, Memoirs of the American Mathematical Society,
vol. 1264, American Mathematical Society, 2019, v+84 pages.
[7] B. Klingler, “Complétude des variétés lorentziennes à courbure constante”, Math. Ann. 306 (1996), no. 2, p. 353-370.
[8] T. Leistner, D. Schliebner, “Completeness of compact Lorentzian manifolds with abelian holonomy”, Math. Ann. 364
(2016), no. 3-4, p. 1469-1503.
[9] J. Marsden, “On completeness of homogeneous pseudo-riemannian manifolds”, Math. J. Indiana Univ. 22 (1973),
p. 1065-1066.
[10] L. Mehidi, A. Zeghib, “On completeness and dynamics of compact Brinkmann spacetimes”, https://arxiv.org/abs/
2205.07243, 2022.
[11] T. J. Munn, “Symmetric spaces, geometric manifolds and geodesic completeness”, 2021, Master’s thesis, University
of Adelaide, School of Mathematical Sciences, https://hdl.handle.net/2440/134331.
[12] B. O’Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics, vol. 103, Academic Press Inc., 1983,
xiii+468 pages.
[13] G. de Rham, “Sur la reductibilité d’un espace de Riemann”, Comment. Math. Helv. 26 (1952), p. 328-344.
[14] A. Romero, M. Sánchez, “New properties and examples of incomplete Lorentzian tori”, J. Math. Phys. 35 (1992), no. 4,
p. 1992-1997.
[15] ——— , “Completeness of compact Lorentz manifolds admitting a timelike conformal Killing vector field”, Proc. Am.
Math. Soc. 123 (1995), no. 9, p. 2831-2833.
[16] H.-H. Wu, “On the de Rham decomposition theorem”, Ill. J. Math. 8 (1964), p. 291-311.


Completeness of certain compact Lorentzian locally symmetric spaces

Adolfo Arroyo-Rabasa

José Simental

An elementary approach to the homological properties of constant-rank operators

Zhong-Xuan Mao

Jing-Feng Tian

Monotonicity and complete monotonicity of some functions involving the modified Bessel functions of the second kind

Ahmad Younso

Comptes Rendus
Mathématique
Ahmad Younso
Consistency of the k-nearest neighbors rule for functional data
Volume 361 (2023), p. 237-242
https://doi.org/10.5802/crmath.402
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 237-242
https://doi.org/10.5802/crmath.402
Statistics / Statistiques
Consistency of the k-nearest neighbors rule
for functional data
Consistance de la règle des k-plus proches voisins pour
des données fonctionnelles
Ahmad Younsoa, b
a MISTEA, Université Montpellier, INRAE, Institut Agro,Montpellier, France
b Department of mathematical statistics, Damascus university, Damascus, Syria
E-mail: ahmad.younso@inrae.fr
Abstract. The problem of nonparametric classification by k-nearest neighbors rule in a general metric space
will be considered. Consistency and strong consistency of the classifier will be established under mild
conditions.
Résumé. Le problème de la classification non paramétrique par la règle des k- plus proches voisins dans un
espace métrique général sera considéré. La consistance et la forte consistance du classifieur seront établies
sous des conditions légères.
2020 Mathematics Subject Classification. 00X99.
Manuscript received 20 April 2021, accepted 26 June 2022.
1. Introduction
Let X be a random variable defined on some probability space (Ω,A ,P) with values in a separable
metric space (F ,ρ) where F is a function space and ρ denotes the metric on F , and let Y be
a random variable with values 0 or 1. In this paper, we study the classical binary supervised
classification model for data from F . Given a new incoming observation X , our goal is to predict
its corresponding label Y . The distribution of the pair (X ,Y ) is well defined by (µ,η) where
µ(B) = P(X ∈ B), for all Borel sets B on F , and η(x) = E(Y |X = x) the regression function of Y
given X = x, for all x ∈ F . In order to predict the unknown label Y of an observation X = x, we
use a classifier that provides a decision rule for this problem. Formally, a classifier is a measurable
mapping g : F −→ {0,1}. Given a classifier g , its corresponding miss-classification error is given
by L = L(g ) = P{g (X ) 6= Y }. In practice, the best classifier is that associated with the smallest
possible error. It is well known that the Bayes classifier given by
g∗(x) =1{η(x)≥1/2},
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
238 Ahmad Younso
where 1A denotes the indicator function of the set A, leads to the lowest possible miss-
classification error, i.e.,
L∗ = L(g∗) = inf
g :F→{0,1}
P{g (X ) 6= Y }.
Unfortunately, g∗ is not available since it depends on the distribution of (X ,Y ) which is generally
unknown. But it is often possible to construct a classifier from a set of n independent and
identically distributed copies Dn = {(X1,Y1), . . . , (Xn ,Yn)} of (X ,Y ). The set Dn is called the
training data. Among the various ways to define a classifier from a training data, one of the most
wide spread and simplest is the k-nearest neighbors (k-N N ) classifier given by
gn(x) =
{
0 if
∑n
i=1 wni Yi ≤ 1/2
1 otherwise,
(1)
where wni = wni (x;Dn) is 1/k if Xi is one of the k-nearest neighbor of x in Dn with respect to the
metric ρ, and wni is zero otherwise with k = k(n) is a sequence of positive integers satisfying
k −→∞ and k/n −→ 0 as n →∞. (2)
Ties are broken by preferring points earlier in the sequence. According to the rule (1), an unclas-
sified element is assigned to the class represented by a majority of its k-nearest neighbors in the
training set. If we let ηn(x) = ∑ni=1 wni Yi be the k-nearest neighbor estimator of η(x), the classi-
fier (1) can be re-written as follows
gn(x) =
{
0 if ηn(x) ≤ 1/2
1 otherwise.
Let Ln = L(gn) = P{gn(X ) 6= Y } be the miss-classification error of gn(x). The main challenge in
this supervised classification setting is to construct a classifier gn whose miss-classification error
will be as close as possible to L∗. In this context, the classifier gn(x) is called consistent if
ELn −→ L∗ as n →∞
and called strongly consistent if
Ln −→ L∗ with probability one as n →∞.
A classifier can be consistent for certain class of distribution of (X ,Y ), but not be consistent for
others. The classifier gn(x) is called (strongly) universally consistent, if it is (strongly) consistent
for all distribution of (X ,Y ). In finite-dimensional spaces, the k-nearest neighbors rule is univer-
sally strongly consistent under classical conditions. (see [5] and [9]). [2] prove the consistency of
the k-N N rule in separable Hilbert spaces. [3] give some examples showing that the results of [6]
on the consistency are no more valid in a general functional metric space (F ,ρ) and they estab-
lish consistency of the k-N N rule on a separable metric space. [1] show that the moving window
rule is not consistent in general metric spaces and give conditions on the space and the regres-
sion function to ensure the (strong) consistency of the estimator. More recently, [7] extend the
Stone’s seminal result to the case of metric spaces when the probability measure of the explana-
tory variables is tight. Then, under slight variations on the hypotheses, they extend the result to
some general metric measure spaces. In this paper, the consistency of the classifier (1) will be
proved under weaker assumptions than that of [3] and [7]. Furthermore, the strong consistency
of the classifier (1) will also be established under some weak conditions. Denote Bx,ε the closed
ball of radius ε > 0 and center at x ∈ F . To establish the main results, we will need the following
Besicovitch condition, for every ε> 0,
lim
δ→0
µ
{
x ∈F :
∣∣∣∣ 1µ(Bx,δ)
∫
Bx,δ
ηdµ−η(x)
∣∣∣∣> ε}= 0. (3)
Ahmad Younso 239
Note that (3) a classical assumption for this kind of results and it holds for example if η(x) is µ-
continuous (for more detail on this topic, see [3]). Besicovitch condition plays an important role
also in the consistency of kernel rules (see [1]). In finite dimension, (3) holds automatically for any
integrable function since it is just the differentiation theorem with respect to a finite measure.
2. Main results
Before we state the main results of this paper, we introduce the following lemma which is needed
in the proofs. Denote
supp(µ) = {x ∈F :µ(Bx,ε) > 0,∀ ε> 0}.
Lemma 1. If (2) is fulfilled then, for each x ∈ supp(µ), there exists a sequence an = an(x) → 0 as
n →∞ such that µ(Bx,an ) ≥
p
k/n.
For the proof of Lemma 1, the reader is referred to [7].
2.1. Consistency of the classifier
The following theorem state the consistency of the classifier (1).
Theorem 2. If (2)–(3) are fulfilled then the k-N N classifier is consistent.
Note that classical condition (2) is the same as that used by [6] in the finite dimensional case
and by [3] in the infinite dimensional case. Condition (3) is used by [3].
Proof. By Theorem 2.2 in [6], whose extension to the infinite dimensional setting is straightfor-
ward, we can write
E(Ln)−L∗ ≤ 2E
∫
F
|η(x)−ηn(x)|µ(dx). (4)
Therefore, the theorem will be proved if we show that
E
∫
F
|η(x)−ηn(x)|µ(dx) −→ 0 as n →∞. (5)
Define
η̂n(x) = 1
nµ(Bx,an )
n∑
i=1
Yi1{Xi∈Bx,an } (6)
with an → 0 is the sequence given in Lemma 1. We have
E
∫
F
|η(x)−ηn(x)|µ(dx) ≤ E
∫
F
|η(x)−Eη̂n(x)|µ(dx)+E
∫
F
|Eη̂n(x)−ηn(x)|µ(dx). (7)
Therefore, it suffices to prove that each term in the right-hand side of (7) tends to zero as n → 0.
Besicovitch condition yields as n → 0,∫
F
|η(x)−Eη̂n(x)|µ(dx) → 0. (8)
It remains to prove that as n → 0,
E
∫
F
|Eη̂n(x)−ηn(x)|µ(dx) → 0. (9)
Clearly, we have
E
∫
F
|Eη̂n(x)−ηn(x)|µ(dx) ≤ E
∫
F
|Eη̂n(x)− η̂n(x)|µ(dx)+E
∫
F
|η̂n(x)−ηn(x)|µ(dx). (10)
240 Ahmad Younso
Therefore, we will prove that each term in the right-hand side of the above inequality tends to
zero. Cauchy–Schwartz inequality yields
E|η̂n(x)−Eη̂n(x)| ≤ E((η̂n(x)−Eη̂n(x))2)1/2 ≤ (var(η̂n(x)))1/2 ≤
(
E(Y 1{X∈Bx,an })
2
n(µ(Bx,an ))
2
)1/2
.
Since |Y | ≤ 1, we obtain
E|η̂n(x)−Eη̂n(x)| ≤
(
1
nµ(Bx,an )
)1/2
. (11)
Hence, Lemma 1 yields
1
nµ(Bx,an )
≤
p
n/k
n
= 1p
kn
. (12)
By (11)–(12) together with Fubini’s theorem and the dominated convergence theorem, we get as
n →∞,
E
∫
F
|Eη̂n(x)− η̂n(x)|µ(dx) → 0. (13)
Now, let X(k)(x) be the k-nearest neighbor of x in Dn and denote rn = rn(x) = ρ(X(k)(x), x). Clearly,
ηn(x) = 1
k
n∑
i=1
Yi1{Xi∈Bx,rn }.
Then, since |Yi | ≤ 1,
|η̂n(x)−ηn(x)| =
∣∣∣∣∣ 1nµ(Bx,an )
n∑
i=1
Yi1{Xi∈Bx,an } −
1
k
n∑
i=1
Yi1{Xi∈Bx,rn }
∣∣∣∣∣
≤
n∑
i=1
∣∣∣∣ 1nµ(Bx,an )1{Xi∈Bx,an } − 1k 1{Xi∈Bx,rn }
∣∣∣∣
≤
∣∣∣∣∣ 1nµ(Bx,an )
n∑
i=1
1{Xi∈Bx,an } −1
∣∣∣∣∣ := |η̃n(x)−Eη̃n(x)|. (14)
Observe that η̃n(x) = η̂n(x) if we let Yi = 1 for all i = 1, . . . ,n. Consequently, the proof of the limit
E
∫
F
|η̃n(x)−Eη̃n(x)|→ 0 (15)
is the same as that of (13). Finally, by (7)–(10), (13) and (14)–(15) the theorem is proved. 
2.2. Strong consistency of the classifier
The following theorem state the strong consistency of the classifier (1).
Theorem 3. Suppose that (2)–(3) are fulfilled. If in addition k/(logn) → ∞, then the k-N N
classifier is strongly consistent.
Note that the classical condition k/(logn) → ∞, used by [4], is crucial to get the strong
consistency of the classifier (2).
Proof. By Theorem 2.2 in [6], the theorem will be proved if we show that∫
F
|η(x)−ηn(x)|µ(dx) −→ 0 as n →∞ with probability one. (16)
Using the term η̂n(x) defined above in (6), we can write∫
F
|η(x)−ηn(x)|µ(dx) ≤
∫
F
|η(x)−Eη̂n(x)|µ(dx)+
∫
F
|Eη̂n(x)−ηn(x)|µ(dx). (17)
As a consequence, by (8), it suffices to show that∫
F
|Eη̂n(x)−ηn(x)|µ(dx) → 0 with probability one as n →∞. (18)
Ahmad Younso 241
Clearly, we have∫
F
|Eη̂n(x)−ηn(x)|µ(dx) ≤
∫
F
|Eη̂n(x)− η̂n(x)|µ(dx)+
∫
F
|η̂n(x)−ηn(x)|µ(dx). (19)
Hence, we will prove each term in the right-hand side of the above inequality tends to zero as
n →∞ with probability one. Let us first deal with the first term in the right-hand side of (19). If
we replace (Xi ,Yi ) by (X ′i ,Y
′
i ), suppose that the value of η̂n(x) is changed to η̂ni (x). Then,∣∣∣∫
F
|Eη̂n(x)− η̂n(x))|µ(dx)−
∫
F
|Eη̂n(x)− η̂ni (x)|µ(dx)
∣∣∣
≤
∫
F
|η̂n(x)− η̂ni (x)|µ(dx) ≤
∫
F
1
nµ(Bx,an )
µ(dx) ≤ 1p
nk
. (20)
The last inequality is due to Lemma 1. Now, for any ε> 0, and n large enough, we have for n large
enough
P
(∫
F
|Eη̂n(x)− η̂n(x)|µ(dx) > ε
)
≤P
(∣∣∣∣∫
F
|Eη̂n(x)− η̂n(x)|µ(dx −
∫
F
|Eη̂n(x)− η̂n(x)|µ(dx)
∣∣∣∣> ε/2) .
Hence, using McDiarmid’s inequality (see [8]) with taking into account (20), we get for n large
enough
P
(∫
F
|Eη̂n(x)− η̂n(x)|µ(dx) > ε
)
≤ 2exp
(
−ε
2k
2
)
Since k/(logn) →∞ by assumption, Borel–Cantelli lemma yields∫
F
|Eη̂n(x)− η̂n(x)|µ(dx) → 0 with probability one as n →∞. (21)
Let us now deal with the second term in the right-hand side of (19). By (14),
|η̂n(x)−ηn(x)| ≤ |η̃n(x)−Eη̃n(x)|, (22)
with η̃n(x) = η̂n(x) if we let Yi = 1 for all i = 1, . . . ,n. Hence, if we follow the same arguments that
use to prove (21), we get∫
F
|η̃n(x)−Eη̃n(x)|µ(dx) → 0 with probability one as n →∞. (23)
Finally, by (8), (17), (19) and (21)–(23) we get (16) and the proof is completed. 
Conclusion. The consistency result of Theorem 2 is obtained under weaker assumptions than
that of [7]. If we let (F ,ρ) = (Rd ,‖·‖) with ‖·‖ denotes the Euclidean norm, then the Stone’s
theorem is considered as a particular case of Theorem 2. Furthermore, Theorem 3 is an extension
of Theorem 11.1 of [6] to a general metric space. In the latter result the authors suppose that µ is
absolutely continuous while µ in Theorem 2 is any probability measure.
References
[1] C. Abraham, G. Biau, B. Cadre, “On the kernel rule for function classification”, Ann. Inst. Stat. Math. 58.
[2] G. Biau, F. Bunea, M. H. Wegkamp, “Functional classification in Hilbert spaces”, IEEE Trans. Inf. Theory 51 (2005),
no. 6, p. 2163-2172.
[3] F. Cérou, A. Guyader, “Nearest neighbor classification in infinite dimension”, ESAIM, Probab. Stat. 10 (2006), p. 340-
355.
[4] K. Chaudhuri, S. Dasgupta, “Rates of Convergence for Nearest Neighbor Classification”, 2014, https://arxiv.org/abs/
1407.0067.
[5] L. Devroye, L. Györfi, A. Krzyżak, G. Lugosi, “On the Strong Universal Consistency of Nearest Neighbor Regression
Function Estimates”, Ann. Stat. 22 (1994), no. 3, p. 1371-1385.
[6] L. Devroye, L. Györfi, G. Lugosi, A probabilistic Theory of Pattern Recognition, Applications of Mathematics, vol. 31,
Springer, 1996.
242 Ahmad Younso
[7] L. Forzani, R. Fraiman, P. Llop, “Consistent Nonparametric Regression for Functional Data Under the
Stone–Besicovitch Conditions”, IEEE Trans. Inf. Theory 58 (2012), no. 11, p. 6697-6708.
[8] C. McDiarmid, “On the method of bounded differences”, in Surveys of combinatorics, London Mathematical Society
Lecture Note Series, vol. 141, Cambridge University Press, 1989, p. 148-188.
[9] C. J. Stone, “Consistent nonparametric regression”, Ann. Stat. 5 (1977), no. 4, p. 595-620.


Consistency of the $k$-nearest neighbors rule for functional data

Xiaobiao Jia

Xuemei Li

Asymptotic behavior of solutions of Monge–Ampère equations with general perturbations of boundary values

Dragan Stankov

The number of nonunimodular roots of a reciprocal polynomial

David Conlon

Yu-Han Wu

Comptes Rendus
Mathématique
David Conlon and Yu-Han Wu
More on lines in Euclidean Ramsey theory
Volume 361 (2023), p. 897-901
Published online: 18 July 2023
https://doi.org/10.5802/crmath.452
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 897-901
https://doi.org/10.5802/crmath.452
Combinatorics / Combinatoire
More on lines in Euclidean Ramsey theory
David Conlon∗, a and Yu-HanWub
a Department of Mathematics, California Institute of Technology, Pasadena, CA
91125, USA
b École Normale Supérieure - PSL, Paris, France
E-mails: dconlon@caltech.edu, yu-han.wu@ens.psl.eu
Abstract. Let ℓm be a sequence of m points on a line with consecutive points at distance one. Answering a
question raised by Fox and the first author and independently by Arman and Tsaturian, we show that there is
a natural number m and a red/blue-colouring of En for every n that contains no red copy of ℓ3 and no blue
copy of ℓm .
Funding. Research supported by NSF Award DMS-2054452.
Manuscript received 29 August 2022, accepted 5 December 2022.
1. Introduction
Let En denote n-dimensional Euclidean space, that is, Rn equipped with the Euclidean metric.
Given two sets X1, X2 ⊂ En , we write En → (X1, X2) if every red/blue-coloring of En contains
either a red copy of X1 or a blue copy of X2, where a copy for us will always mean an isometric
copy. Conversely, En ↛ (X1, X2) means that there is some red/blue-coloring of En which contains
neither a red copy of X1 nor a blue copy of X2.
The study of which sets X1, X2 ⊂ En satisfy En → (X1, X2) is a particular case of the Euclidean
Ramsey problem, which has a long history going back to a series of seminal papers [6–8] of Erdős,
Graham, Montgomery, Rothschild, Spencer and Straus in the 1970s. Despite the vintage of the
problem, surprisingly little progress has been made since these foundational papers (though
see [9, 12] for some important positive results). For instance, it is an open problem, going back
to the papers of Erdős et al. [7], as to whether, for every n, there is m such that En ↛ (X , X ) for
every X ⊂ En with |X | = m.
Write ℓm for the set consisting of m points on a line with consecutive points at distance
one. Perhaps because it is a little more accessible than the general problem, the question of
determining which n and X satisfy the relation En → (ℓ2, X ) has received considerable attention.
For instance, it is known [11, 14] that E2 → (ℓ2, X ) for every four-point set X ⊂ E2 and that
E2 → (ℓ2,ℓ5). On the other hand [5], there is a set X of 8 points in the plane, namely, a regular
heptagon with its center, such that E2 ↛ (ℓ2, X ).
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
898 David Conlon and Yu-Han Wu
In higher dimensions, by combining results of Szlam [13] and Frankl and Wilson [10], it was
observed by Fox and the first author [4] that En → (ℓ2,ℓm) provided m ≤ 2cn for some positive
constant c (see also [1, 2] for some better bounds in low dimensions). Our concern here will
be with a question raised independently by Fox and the first author [4] and also by Arman and
Tsaturian [1], namely, as to whether an analogous result holds with ℓ2 replaced by ℓ3. That is, for
every natural number m, is there a natural number n such that En → (ℓ3,ℓm)? We answer this
question in the negative.
Theorem 1. There exists a natural number m such that En ↛ (ℓ3,ℓm) for all n.
Before our work, the best result that was known in this direction was a 50-year-old result of
Erdős et al. [6], who showed that En ↛ (ℓ6,ℓ6) for all n. Their proof uses a spherical colouring,
where all points at the same distance from the origin receive the same colour. We will also use a
spherical colouring, though, unlike the colouring in [6], which is entirely explicit, our colouring
will be partly random.
2. Preliminaries
In this short section, we note two key lemmas that will be needed in our proof. The first says that
certain real-valued quadratic polynomials are reasonably well-distributed modulo a prime q .
Lemma 2. Let p(x) = x2 + αx + β, where α and β are real numbers, and let q be a prime
number. Then, for m = q3, the set {p(i )}mi=1 overlaps with at least q/6 of the intervals [ j , j +1) with
0 ≤ j ≤ q −1 when considered mod q.
Proof. By a standard argument using the pigeonhole principle, there exists some k ≤ q2 such
that |kα| ≤ 1/q mod q . We split into two cases, depending on whether k is a multiple of q or not.
Suppose first that k ̸≡ 0 mod q and consider the set of values {p(ki )}qi=1. Note first that {i 2}
q
i=1
is a set of (q +1)/2 distinct integers mod q , so, since k is not a multiple of q , the same is also true
of the set {k2i 2}qi=1. Hence, letting p1(x) = x2 +β, we see that the set {p1(ki )}
q
i=1 overlaps with at
least q/2 of the intervals [ j , j +1) with 0 ≤ j ≤ q −1 when considered mod q . But |kiα| ≤ 1 mod q
for all 1 ≤ i ≤ q , so that |p(ki )−p1(ki )| ≤ 1 for all 1 ≤ i ≤ q . Therefore, since exactly three different
intervals are within distance one of any particular interval, the set {p(ki )}qi=1 overlaps with at least
q/6 of the intervals [ j , j +1) mod q .
Suppose now that k = sq for some s ≤ q . Then sqα= r q + ϵ for some |ϵ| ≤ 1/q , which implies
that α = rs + ϵ′, where |ϵ′| ≤ 1/q2. Without loss of generality, we may assume that r and s have
no common factors. Consider now the polynomial p2(x) = x2 + rs x and the set {p2(si )}
q
i=1. Since
p2(si ) = s2i 2+r i , it is easy to check that p2(si ) ≡ p2(s j ) mod q if and only if s2(i+ j )+r ≡ 0 mod q .
Since r and s are coprime, this implies that the set {p2(si )}
q
i=1 takes at least q/2 values mod q .
Hence, letting p3(x) = x2 + rs x +β, we see that the set {p3(si )}
q
i=1 overlaps with at least q/2 of the
intervals [ j , j +1) with 0 ≤ j ≤ q −1 when considered mod q . But, since |α− r /s| ≤ 1/q2, we have
that |p(si )−p3(si )| = |α− rs |si ≤ 1, so that, as above, the set {p(si )}
q
i=1 overlaps with at least q/6
of the intervals [ j , j +1) mod q . □
Given M real polynomials p1, . . . , pM in N variables, a vector σ ∈ {−1,0,1}M is called a sign
pattern of p1, . . . , pM if there exists some x ∈ RN such that the sign of pi (x) is σi for all 1 ≤ i ≤ M .
The second result we need is the Oleinik–Petrovsky–Thom–Milnor theorem (see, for example, [3]),
which, for N fixed, gives a polynomial bound for the number of sign patterns.
Lemma 3. For M ≥ N ≥ 2, the number of sign patterns of M real polynomials in N variables, each
of degree at most D, is at most
( 50DM
N
)N
.
David Conlon and Yu-Han Wu 899
3. Proof of Theorem 1
Suppose that a1, a2, a3 ∈ Rn form a copy of ℓ3 with |a1 − a2| = |a2 − a3| = 1. If the points are at
distances x1, x2 and x3, respectively, from the origin o and the angle a1a2o is θ, then we have
x21 = x22 +1−2x2 cosθ
and
x23 = x22 +1+2x2 cosθ.
Adding the two gives
x21 +x23 = 2x22 +2.
Similarly, if a1, a2, . . . , am ∈Rn form a copy of ℓm with |ai −ai+1| = 1 for all i = 1,2, . . . ,m −1, then,
again writing xi for the distance of ai from the origin, we have
x2i−1 +x2i+1 = 2x2i +2
for all i = 2, . . . ,m − 1. Given these observations, our aim will be to colour R≥0 so that there is
no red solution to y1 + y3 = 2y2 +2 and no blue solution to the system yi−1 + yi+1 = 2yi +2 with
i = 2, . . . ,m−1. Assuming that we have such a colouring χ, we can simply colour a point a ∈Rn by
χ(|a|2) and it is easy to check that there is no red copy of ℓ3 and no blue copy of ℓm .
We have therefore moved our problem to one of finding a natural number m and a colouring
χ of R≥0 with no red solution to y1 + y3 = 2y2 +2 and no blue solution to the system yi−1 + yi+1 =
2yi +2 with i = 2, . . . ,m−1. Let q be a prime number. We will take m = q3 and defineχ by choosing
an appropriate colouring χ′ of Zq and then setting χ(y) = χ′(⌊y⌋ mod q) for all y ∈ R≥0. Our aim
now is to show that there is a suitable choice for χ′. For this, we consider a random red/blue-
colouring χ′ of Zq and show that, for q sufficiently large, the probability that χ contains either of
the banned configurations is small.
Concretely, suppose that Zq is coloured randomly in red and blue with each element of Zq
coloured red with probability p = q−3/4 and blue with probability 1− p. With this choice, the
expected number of solutions in red to any of the equations y1 + y3 = 2y2 + c with c ∈ {1,2,3} is at
most
3p3q2 +9p2q < 12q−1/4 < 1
2
,
where we used that there are at most 3q solutions to any of our 3 equations with two of the
variables {y1, y2, y3} being equal and that q is sufficiently large. Note that if there are indeed no
red solutions to these three equations over Zq , then there is no red solution to y1 + y3 = 2y2 +2 in
the colouring χ of R. Indeed, if yi = ni +ϵi with 0 ≤ ϵi < 1, then ni is coloured red in χ′ and
n1 +n3 = 2n2 +2+2ϵ2 −ϵ1 −ϵ3.
But |2ϵ2 −ϵ1 −ϵ3| < 2, so we must have
n1 +n3 = 2n2 + c
for c ∈ {1,2,3}. However, we know that there are no red solutions to any of these equations in the
colouring χ′, so there is no red solution to y1 + y3 = 2y2 +2 in the colouring χ.
For the blue configurations, we first observe that if the yi satisfy the equations yi−1 + yi+1 =
2yi + 2 with i = 2, . . . ,m − 1 with y1 = a and y2 = a +d , then yi = a + (i − 1)d + (i 2 − 3i + 2). In
particular, by Lemma 2, at least q/6 elements of the sequence y1, . . . , ym lie in different intervals
[ j , j +1) with 0 ≤ j ≤ q −1 when considered mod q .
Our aim now is to apply Lemma 3 to count the number of different ways in which a set of
solutions (y1, y2, . . . , ym) to our system of equations can overlap the collection of intervals [ j , j +1)
mod q . Without loss of generality, we may assume that 0 ≤ a,d < q . Since, under this assumption,
any set of solutions over R to our system of equations is contained in the interval [0,2m2), it will
suffice to count the number of feasible overlaps with the intervals [ j , j +1) with 0 ≤ j ≤ 2m2 −1.
900 David Conlon and Yu-Han Wu
Since we need to check at most two linear inequalities in the two variables a and d to check
whether each of the m points are placed in each of the 2m2 intervals, we can apply Lemma 3 with
N = 2, D = 1 and M = 2 ·m ·2m2 = 4m3 to conclude that the points y1, . . . , ym overlap the intervals
[ j , j +1) with 0 ≤ j ≤ 2m2 −1 in at most (100m3)2 = 104m6 different ways. But now, since at least
q/6 of the yi must always be in distinct intervals, a union bound implies that the probability we
have a blue solution to our system of equations is at most
104m6(1−q−3/4)q/6 < 1
2
for m sufficiently large. Combined with our earlier estimate for the probability of a red solution to
y1+ y3 = 2y2+2, we see that for m sufficiently large (m = 1050 will suffice) there exists a colouring
with no red ℓ3 and no blue ℓm , as required.
4. Concluding remarks
We say that a set X ⊂ Ed is Ramsey if for every natural number r there exists n such that every
r -colouring of En contains a monochromatic copy of X . In [4], it was shown that a set X is
Ramsey if and only if for every natural number m and every fixed K ⊂ Em there exists n such
that En → (X ,K ). We suspect that there may be an even simpler characterisation.
Conjecture 4. A set X is Ramsey if and only if for every natural number m there exists n such that
En → (X ,ℓm).
Of course, by the result mentioned above, we already know that if X is Ramsey, then En →
(X ,ℓm) for n sufficiently large. It therefore remains to show that if X is not Ramsey, then there
exists m such that En ↛ (X ,ℓm) for all n. To prove this in full generality might be difficult.
However, an important result of Erdős et al. [6] says that if X is Ramsey, then it must be spherical,
in the sense that it must be contained in the surface of a sphere of some dimension. Thus, a first
step towards Conjecture 4 might be to prove the following.
Conjecture 5. For every non-spherical set X , there exists a natural number m such that En ↛
(X ,ℓm) for all n.
The simplest example of a non-spherical set is the line ℓ3, so our main result may be seen as a
verification of Conjecture 5 in this particular case. The next case of interest seems to be when X
consists of three points a1, a2, a3 on a line, but now with |a1 − a2| = 1 and |a2 − a3| = α for some
irrational α.
References
[1] A. Arman, S. Tsaturian, “Equally spaced collinear points in Euclidean Ramsey theory”, https://arxiv.org/abs/1705.
04640, 2017.
[2] ——— , “A result in asymmetric Euclidean Ramsey theory”, Discrete Math. 341 (2018), no. 5, p. 1502-1508.
[3] S. Basu, R. Pollack, M.-F. Roy, Algorithms in Real Algebraic Geometry, 2nd ed., Algorithms and Computation in
Mathematics, vol. 10, Springer, 2006.
[4] D. Conlon, J. Fox, “Lines in Euclidean Ramsey theory”, Discrete Comput. Geom. 61 (2019), no. 1, p. 218-225.
[5] G. Csizmadia, G. Tóth, “Note on a Ramsey-type problem in geometry”, J. Comb. Theory, Ser. A 65 (1994), no. 2, p. 302-
306.
[6] P. Erdős, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, E. G. Straus, “Euclidean Ramsey theorems I”, J.
Comb. Theory, Ser. A 14 (1973), p. 341-363.
[7] ——— , “Euclidean Ramsey theorems II”, in Infinite and finite sets (Colloq., Keszthely, 1973), Vol. I, Colloquia
Mathematica Societatis János Bolyai, vol. 529, North-Holland, 1973, p. 529-557.
[8] ——— , “Euclidean Ramsey theorems III”, in Infinite and finite sets (Colloq., Keszthely, 1973), Vol. I, Colloquia
Mathematica Societatis János Bolyai, vol. 529, North-Holland, 1973, p. 559-583.
David Conlon and Yu-Han Wu 901
[9] P. Frankl, V. Rödl, “A partition property of simplices in Euclidean space”, J. Am. Math. Soc. 3 (1990), no. 1, p. 1-7.
[10] P. Frankl, R. M. Wilson, “Intersection theorems with geometric consequences”, Combinatorica 1 (1981), p. 357-368.
[11] R. Juhasz, “Ramsey type theorems in the plane”, J. Comb. Theory, Ser. A 27 (1979), p. 152-160.
[12] I. Kříž, “Permutation groups in Euclidean Ramsey Theory”, Proc. Am. Math. Soc. 112 (1991), no. 3, p. 899-907.
[13] A. D. Szlam, “Monochromatic translates of configurations in the plane”, J. Comb. Theory, Ser. A 93 (2001), no. 1,
p. 173-176.
[14] S. Tsaturian, “A Euclidean Ramsey result in the plane”, Electron. J. Comb. 24 (2017), no. 4, article no. P4.35 (9 pages).


More on lines in Euclidean Ramsey theory

Jean B. Lasserre

Pell’s equation, sum-of-squares and equilibrium measures on a compact set

Faouaz Saadi

Radouan Daher

Equivalence of K-functionals and modulus of smoothness generated by a Dunkl type operator on the interval $(-1, 1)$

Jeffrey S. Case

A simple construction of the Rumin algebra

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