and Tiling Problems
Jan Corsten
A thesis submitted for the degree of
Doctor of Philosophy
Department of Mathematics
The London School of Economics
and Political Science
I certify that the thesis I have presented for examination for the PhD degree of the London School of Economics and Political Science is solely my own work, with the exceptions outlined below.
The copyright of this thesis rests with the author. Quotation from it is permitted, provided that full acknowledgement is made. In accordance with the regulations, I have deposited an electronic copy of it in LSE Theses Online held by the British Library of Political and Economic Science and have granted permission for my thesis to be made available for public reference. Otherwise, this thesis may not be reproduced without my prior written consent. I warrant that this authorisation does not, to the best of my belief, infringe the rights of any third party.
I declare that this thesis consists of 46960 words. Statement of co-authored work
The contents of Chapter2are well-known results of various authors not including myself.
I confirm that Chapter3contains the following joint work. Section 3.1is based on [18], which is jointly co-authored with Sebastián Bustamante, Nóra Frankl, Alexey Pokrovskiy and Jozef Skokan. Section3.2is based on [31], which is jointly co-authored with Walner Mendonça.
I confirm that Chapter4contains the following joint work. Section 4.1is based on [30], which is jointly co-authored with Louis DeBiasio, Ander Lamaison and Richard Lang. Section4.2is based on [29], which is jointly co-authored with Louis DeBiasio and Paul McKenney.
I confirm that Chapter5is based on [3], which is jointly co-authored with Peter Allen, Julia Böttcher, Ewan Davies, Matthew Jenssen, Patrick Morris, Barnaby Roberts and Jozef Skokan.
In this thesis, we study a variety of different extremal graph colouring and tiling problems in finite and infinite graphs.
Confirming a conjecture of Gyárfás, we show that for all 𝑘 , 𝑟 ∈ N there is a
constant𝐶 >0 such that the vertices of every𝑟-edge-coloured complete𝑘-uniform
hypergraph can be partitioned into a collection of at most𝐶 monochromatic tight
cycles. We shall say that the family of tight cycles hasfinite𝑟-colour tiling number.
We further prove that, for all natural numbers𝑘, 𝑝and𝑟, the family of 𝑝-th powers
of 𝑘-uniform tight cycles has finite𝑟-colour tiling number. The case where 𝑘 = 2
settles a problem of Elekes, Soukup, Soukup and Szentmiklóssy. We then show that for all natural numbers Δ, 𝑟, every family F = {𝐹1, 𝐹2, . . .} of graphs with 𝑣(𝐹𝑛) =𝑛 andΔ(𝐹𝑛) ≤ Δfor every𝑛 ∈ Nhas finite𝑟-colour tiling number. This makes progress on a conjecture of Grinshpun and Sárközy.
We study Ramsey problems for infinite graphs and prove that in every 2-edge-colouring of𝐾
N, the countably infinite complete graph, there exists a monochromatic infinite path𝑃such that𝑉(𝑃)has upper density at least(12+
√
8)/17≈0.87226 and
further show that this is best possible. This settles a problem of Erdős and Galvin. We study similar problems for many other graphs including trees and graphs of bounded degree or degeneracy and prove analogues of many results concerning graphs with linear Ramsey number in finite Ramsey theory.
We also study a different sort of tiling problem which combines classical problems from extremal and probabilistic graph theory, the Corrádi–Hajnal theorem and (a special case of) the Johansson–Kahn–Vu theorem. We prove that there is some constant𝐶 > 0 such that the following is true for every 𝑛 ∈ 3N and every 𝑝 ≥ 𝐶 𝑛−2/3(log𝑛)1/3. If𝐺 is a graph on𝑛vertices with minimum degree at least 2𝑛/3,
then𝐺𝑝 (the random subgraph of𝐺 obtained by keeping every edge independently
Firstly, I would like to thank my supervisors Peter and Julia, and Jozef, who acted like a third supervisor. I would like to express my gratitude for their guidance throughout my PhD, for all the discussions (mathematical and non-mathematical), and for encouraging me to participate in workshops, conferences and summer schools around the world.
I would like to thank my fellow PhD students and good friends Attila and Nóra for our collaboration and for their unlimited supply of fun puzzles. I would like also like to thank all other PhD students for making the PhD office a great work environment, and for all the fun times we had at LSE and around the world. I would like to thank Kate, Enfale, Becca, Sarah and Ed for their administrative support and creating such a nice atmosphere in the department. I would like to thank Jozef and Jan for keeping us well hydrated.
Finally, I would like to thank my parents and my sister for their constant support throughout my life.
1 Introduction 8 1.1 Notation . . . 10 1.1.1 Elementary Notation . . . 10 1.1.2 O-Notation . . . 11 1.1.3 Graphs . . . 12 1.1.4 Hypergraphs . . . 14
1.1.5 Edge Colourings and Ramsey Numbers . . . 17
1.1.6 Infinite Graph Theory . . . 17
1.1.7 Random Graphs . . . 18
1.2 Graphs with Linear Ramsey Number . . . 20
1.2.1 Paths and Cycles . . . 21
1.2.2 Trees . . . 22
1.2.3 Graphs with Bounded Degree . . . 22
1.3 Monochromatic Graph Tiling Problems . . . 24
1.3.1 History . . . 24
1.3.2 New Results . . . 25
1.4 Ramsey Problems for Infinite Graphs . . . 31
1.4.1 Infinite Paths . . . 31
1.4.2 Infinite Trees . . . 33
1.4.3 Infinite Graphs with “Linear Ramsey Number” . . . 34
1.5 Robust Triangle Tilings in Random Graphs . . . 38
1.5.1 Shamir’s Problem . . . 38
2 Preliminaries 42
2.1 The Absorption Method. . . 42
2.2 Graph Regularity . . . 44
2.2.1 Definitions and Basic Properties . . . 44
2.2.2 The Regularity Lemma . . . 49
2.2.3 Finding Large Regular Cylinders . . . 51
2.2.4 The Blow-up Lemma . . . 53
2.3 Hypergraph Regularity . . . 55
2.4 Probabilistic Tools . . . 59
2.5 Entropy . . . 60
3 Monochromatic Graph Tiling Problems 67 3.1 Tiling Coloured Hypergraphs with Tight Cycles . . . 67
3.1.1 Overview . . . 67
3.1.2 Absorption Method for Hypergraphs . . . 68
3.1.3 Absorption Lemma . . . 70
3.1.4 Proof of Theorem 1.3.10. . . 75
3.2 Tiling Coloured Graphs with Graphs of Bounded Degree . . . 78
3.2.1 Overview . . . 78
3.2.2 Tools for the Absorption Method . . . 80
3.2.3 The Absorption Lemma . . . 81
3.2.4 Proof of Theorem 1.3.16 . . . 88
3.2.5 Concluding Remarks . . . 93
4 Ramsey Problems for Infinite Graphs 95 4.1 Ramsey Upper Density of Paths . . . 95
4.1.1 Overview . . . 95
4.1.2 Proof of Theorem 1.4.4 . . . 96
4.1.3 Proof of Theorem 1.4.5 . . . 99
4.1.4 From Path Forests to Paths . . . 100
4.1.5 From Simple Forests to Path Forests . . . 102
4.1.6 Upper Density of Simple Forests . . . 105
4.2 Upper Density of Monochromatic Subgraphs . . . 114
4.2.1 Overview . . . 114
4.2.2 Preliminaries . . . 116
4.2.3 Constructions . . . 118
4.2.4 Ultrafilters and Embedding . . . 121
4.2.5 Bipartite Ramsey Densities. . . 124
4.2.6 Trees . . . 126
4.2.7 Graphs of Bounded Chromatic Number . . . 131
4.2.8 Graphs of Bounded Ruling Number . . . 133
5 Robust Triangle Tilings 136 5.1 Overview . . . 136
5.1.1 Set-up . . . 136
5.1.2 Proof Outline . . . 139
5.2 Counting Triangles inΓ𝑝 . . . 140
5.3 Embedding (Partial) Triangle Tilings . . . 147
5.3.1 Counting Almost Triangle Tilings . . . 147
5.3.2 Extending Almost Triangle Tilings. . . 148
5.3.3 Completing Triangle Tilings . . . 153
5.4 Proof of the Local Distribution Lemma. . . 153
5.4.1 A Simplification . . . 153
5.4.2 Entropy lemma . . . 155
5.4.3 Counting via Comparison . . . 165
5.5 Reducing to Regular Triples . . . 169
5.5.1 Preparation . . . 169
1
Introduction
Extremal graph theory seeks to answer questions of the following form: what is the largest or smallest value of some graph parameter among all graphs of a given class? The possibly first result in this area determines how many edges a triangle-free graph can have.
Theorem 1.0.1 (Mantel [94]). Every graph with 𝑛 vertices and more than 𝑛2/4
edges contains a triangle. Furthermore, the complete bipartite graph with parts of sized𝑛/2e andb𝑛/2ccontainsb𝑛2/4c edges and is triangle-free.
Another classical example of extremal graph theory is the study of Ramsey num-bers. TheRamsey numberof a graph𝐻, denoted by R(𝐻), is the smallest integer𝑁
such that in every two-colouring of the edges of the complete graph on 𝑁 vertices 𝐾𝑁, we can find a monochromatic copy of 𝐻 (that is a subgraph of 𝐾𝑁 which is isomorphic to 𝐻 and all its edges receive the same colour). Ramsey numbers are
named after Ramsey, who proved the following result.
Theorem 1.0.2(Ramsey [100]). R(𝐾𝑛) < ∞for every𝑛 ∈N.
It immediately follows from Ramsey’s theorem that Ramsey numbers of all finite graphs are finite and there has been a lot of work in determining and estimating them. One topic of interest is to classify which graphs have small, i.e. linear, Ramsey number. To be more precise, define a sequence of graphs to be a family of graphs F = {𝐹1, 𝐹2, . . .} where 𝐹𝑛 has 𝑛 vertices for every 𝑛 ∈ N. Such F is said to have linear Ramsey number if there exists a constant 𝐶 > 0 such that
it follows directly from the definition that if F has linear Ramsey number, it is
possible to cover a linearly large part of every 2-edge-coloured 𝐾𝑛 using only one
monochromatic copy fromF.
Observation 1.0.3. For every sequence of graphs F with linear Ramsey number, there is a constant𝑐 =𝑐(F ) > 0, such that every two-edge-coloured𝐾𝑛contains a
monochromatic copy of some𝐹 ∈ F with at least𝑐𝑛vertices.
By greedily applying Observation1.0.3to the set of remaining vertices, we can actually cover most of𝐾𝑛using only “few” monochromatic copies fromF.1 Observation 1.0.4. For every 𝜀 > 0 and every sequence of graphsF with linear
Ramsey number, there is a constant 𝐶 = 𝐶(F, 𝜀) such that the following is true
for every two-edge-coloured 𝐾𝑛. There is a collection of at most𝐶 vertex-disjoint
monochromatic copies fromF whose union covers at least(1−𝜀)𝑛vertices.
In other words, Observation 1.0.4 asserts that it is possible to cover almost all vertices of every two-edge coloured 𝐾𝑛 by vertex-disjoint monochromatic copies
fromF whose average size is linearly large. It is natural to ask for which families F we can extend Observation 1.0.4to cover all the vertices of𝐾𝑛. Such problems are calledgraph tiling problemsand we shall discuss some of them in Chapter3.
Ramsey’s theorem further implies that every two-edge-coloured𝐾
N(the countably infinite complete graph) contains a monochromatic copy of 𝐾N.2 Therefore, there
is no direct extensions of Ramsey numbers to infinite graphs. It is possible however to generalise the concept of graphs with linear Ramsey number to infinite graphs using a density formulation similar as in Observation1.0.3. Theupper densityof a set 𝐴 ⊆ Nis defined as d(𝐴) :=lim sup
𝑛→∞|𝐴∩ [𝑛] | /𝑛. It is not hard to see that there exist 2-edge-colourings of𝐾
Nsuch that the vertex-set of every monochromatic copy of 𝐾
N has upper density 0 (e.g., a random colouring). In Chapter4 we will discuss infinite graphs for which this is not the case and prove infinite analogues of many results about graphs with linear Ramsey number.
1This was first observed by Erdős, Gyárfás and Pyber [42] to the best of the author’s knowledge. 2In fact, this is how the theorem was originally stated.
We will also study a different sort of graph tiling problems in random graphs. The binomial random graph𝐺(𝑛, 𝑝) is the random subgraph of𝐾𝑛in which every
edge of 𝐾𝑛is present independently with probably 𝑝. This random graph was first
introduced by Gilbert [56], while Erdős and Rényi [44] independently introduced the related random graph 𝐺(𝑛, 𝑚). Erdős and Rényi then initiated the systematic
study of both types of random graphs in a series of papers [45, 46, 47, 48] and it has been a very active research topic ever since. One of the central questions in the theory of random graphs is to determine threshold probabilities for certain graph properties. A classical example of this is the following theorem, which determines when𝐺(𝑛, 𝑝)is connected.
Theorem 1.0.5(Gilbert [56]). LetC𝑛 denote the set of all connected graphs on 𝑛 vertices. Then, for every𝜀 > 0, we have
lim 𝑛→∞P[ 𝐺(𝑛, 𝑝(𝑛)) ∈ C𝑛] =
0 if 𝑝(𝑛) ≤ (1−𝜀)log(𝑛)/𝑛for all𝑛∈N,
1 if 𝑝(𝑛) ≥ (1+𝜀)log(𝑛)/𝑛for all𝑛∈N.
We will study similar questions regarding the existence of a triangle tiling in random subgraphs of certain deterministic graphs in Chapter5.
1.1 Notation
1.1.1 Elementary Notation
We denote byRthe set of real numbers, byR≥0the set of non-negative real numbers and byR+ the set of positive real numbers. We denote byN:={1,2, . . .}the set of
positive integers and byN0 :={0,1,2, . . .}the set of non-negative integers.
Given a set𝑉, we denote by 2𝑉 the set of all subsets of𝑉 and, given 𝑘 ∈ N0,
we denote by 𝑉 𝑘
the set of all 𝑘-element subsets of𝑉. For integers 0 ≤ 𝑡 ≤ 𝑛, we
define𝑛!𝑡 to be the number of ways to select a list of𝑡 distinct numbers from [𝑛]. That is,𝑛!𝑡 :=
𝑛!
(𝑛−𝑡)! =𝑛· (𝑛−1) · · · (𝑛−𝑡+1).
Given 𝑘 ∈ N0, we denote by [𝑘] := {1, . . . , 𝑘} the set of the first 𝑘 positive
non-negative integers. Given𝑘 ≤ ℓ ∈N0, we denote by[𝑘 , ℓ] :={𝑘 , . . . , ℓ} the set
of all integers𝑧with 𝑘 ≤ 𝑧 ≤ ℓ. Given 𝑎 ≤ 𝑏 ∈R, we denote by [𝑎, 𝑏]
R the set of all reals𝑥with𝑎 ≤ 𝑥 ≤ 𝑏. Given𝑎, 𝑏∈ R, we denote by (𝑎, 𝑏) =(𝑎, 𝑏)
Rthe set of all reals𝑥with𝑎 < 𝑥 < 𝑏.
We denote by exp the exponential function with base𝑒 and, for 𝑘 ∈ N, by exp𝑘
the 𝑘-th composition of the exponential function. We denote by log the natural
logarithm, that is, the logarithm with base𝑒.
In this thesis we will frequently break long proofs into smaller claims. We denote bythe end of the proof of such a claim and by the end of the main proof.
1.1.2 O-Notation
Let 𝑓 , 𝑔 :N→Rbe functions taking non-negative values. We say that
• 𝑓 =𝑂(𝑔) if there is some𝐶 >0 such that 𝑓(𝑛) ≤𝐶·𝑔(𝑛)for all sufficiently
large𝑛 ∈N.
• 𝑓 = Ω(𝑔)if𝑔=𝑂(𝑓).
• 𝑓 = Θ(𝑔)if 𝑓 =𝑂(𝑔) and𝑔 =𝑂(𝑓).
• 𝑓 =𝑜(𝑔) if for all𝜀 > 0, there is some𝑛0 > 0 so that 𝑓(𝑛) < 𝜀·𝑔(𝑛)for all 𝑛 ≥ 𝑛0.
• 𝑓 =𝜔(𝑔)if𝑔 =𝑜(𝑓).
If the functions 𝑓 , 𝑔 depend on more than one variable, we shall indicate the
parameter tending to infinity in the subscript and consider all other variables as constant. For example,𝑛2+𝑚3=𝑜𝑛(𝑛3+𝑚2) and𝑛2+𝑚3=𝜔𝑚(𝑛3+𝑚2).
Furthermore, given real numbers𝑎, 𝑏, 𝑐, we say𝑎=𝑏±𝑐if|𝑎−𝑏| ≤ |𝑐|, and we
will frequently use the following slightly informal notation: Given reals𝜀, 𝛿 > 0,
we say𝛿 𝜀if𝛿 can be chosen arbitrarily small in terms of𝜀. This shall replace
the quantification “for all𝜀 >0 there exists some𝛿0> 0 such that for all 0 < 𝛿 ≤ 𝛿0
1.1.3 Graphs
We use standard notation from graph theory. A graph 𝐺 = (𝑉 , 𝐸) is a tuple
consisting of a set𝑉, called the vertices of𝐺, and a set𝐸 ⊆ 𝑉
2
, called theedges of 𝐺. We write 𝑉(𝐺) to refer to the vertex-set of 𝐺 and 𝐸(𝐺) to refer to the
edge-set of𝑉. We further denote the sizes of𝑉(𝐺) and 𝐸(𝐺) by 𝑣(𝐺) := |𝑉(𝐺) |
and𝑒(𝐺):= |𝐸(𝐺) |. Note that𝑉(𝐺)and𝐸(𝐺)are possibly infinite. We will often
abbreviate an edge{𝑢, 𝑣} ∈𝐸(𝐺) as𝑢𝑣.
Definition 1.1.1 (Neighbourhoods and degrees). Given a graph 𝐺, a vertex 𝑣 ∈ 𝑉(𝐺) and a set 𝑈 ⊆ 𝑉(𝐺), we define the neighbourhood of 𝑣 in 𝑈 by 𝑁𝐺(𝑣 , 𝑈) := {𝑢 ∈ 𝑈 : 𝑢𝑣 ∈ 𝐸(𝐺)}. Furthermore, the degree of 𝑣 in 𝑈 is
given by deg𝐺(𝑣 , 𝑈) := |𝑁𝐺(𝑣 , 𝑈) |. Given a set of vertices 𝑉 ⊆ 𝑉(𝐺), we
define the common neighbourhood of 𝑉 in 𝑈 by 𝑁𝐺(𝑉;𝑈) := Ñ𝑣∈𝑉𝑁𝐺(𝑣 , 𝑈)
and deg𝐺(𝑉;𝑈) := |𝑁𝐺(𝑉;𝑈) |. Given two vertices 𝑣1, 𝑣2 ∈ 𝑉(𝐺), we write
𝑁𝐺(𝑣1, 𝑣2;𝑈) for 𝑁𝐺({𝑣1, 𝑣2};𝑈), and similarly we write deg𝐺(𝑣1, 𝑣2;𝑈) for deg𝐺({𝑣1, 𝑣2};𝑈). If 𝑈 = 𝑉(𝐺), we simply write 𝑁𝐺(𝑣), deg𝐺(𝑣), 𝑁𝐺(𝑉) and deg𝐺(𝑉) and if𝐺 is clear from context we drop the subscript.
Definition 1.1.2(Families and sequences of graphs). A family of graphs F is an
infinite set of graphs. Asequence of graphsis a family of graphsF ={𝐹1, 𝐹2, . . .}
with𝑣(𝐹𝑛) =𝑛for all𝑛 ∈N.
Definition 1.1.3 (Maximum and minimum degree). The maximum degree of 𝐺
is defined by Δ(𝐺) := sup𝑣∈𝑉(𝐺)deg(𝑣) and the minimum degree is defined by 𝛿(𝐺) :=min𝑣∈𝑉(𝐺)deg(𝑣). Themaximum degreeof a family of graphsF is given
byΔ(F ) :=sup𝐹∈F Δ(𝐹).
Definition 1.1.4(Degeneracy). A graph𝐺 is called𝑑-degeneratefor some𝑑 ∈Nif
there is a linear order<of𝑉(𝐺)such that|𝑁(𝑣) ∩{𝑢∈𝑉(𝐺) :𝑢 < 𝑣}| ≤ 𝑑for every 𝑣 ∈𝑉(𝐺). We denote by degen(𝐺)the smallest𝑑 ∈Nsuch that𝐺 is𝑑-degenerate.
Thedegeneracyof a family of graphsF is given by degen(F ) :=sup𝐹∈F degen(𝐹).
Note that we have degen(𝐺) ≤Δ(𝐺)and 𝜒(𝐺) ≤ degen(𝐺) +1 for every graph 𝐺. Furthermore, there are families with finite degeneracy but infinite maximum
Definition 1.1.5 (Subgraphs). A graph 𝐻 is called a subgraph of a graph 𝐺 if 𝑉(𝐻) ⊆ 𝑉(𝐺) and 𝐸(𝐻) ⊆ 𝐸(𝐺). We call 𝐻 spanning if 𝑉(𝐻) = 𝑉(𝐺). If 𝑈 ⊆𝑉(𝐺), we define𝐺[𝑈]to be the subgraph of𝐺induced by𝑈, i.e.𝑉(𝐺[𝑈]) =𝑈
and𝐸(𝐺[𝑈]) ={𝑒 ∈𝐸(𝐺) :𝑒 ⊆𝑈}.
Definition 1.1.6(Independent sets). A set𝐼 ⊆ 𝑉(𝐺) is calledindependentif there
is no edge 𝑒 ∈ 𝐸(𝐺) with 𝑒 ⊆ 𝐼. The independence number of 𝐺 is given by 𝛼(𝐺) := sup{|𝐼| : 𝐼 ∈ I (𝐺)}, whereI (𝐺) denotes the set of all independent sets
in𝐺. Given a partition Pof𝑉(𝐺), we say that𝐺 isP-partite if every𝑈 ∈ P is an
independent set. A graph𝐺is𝑡-partite if it isP-partite for some partitionPof𝑉(𝐺)
with𝑡 parts. The minimum 𝑡 ∈ N such that 𝐺 is𝑡-partite is called the chromatic
numberof𝐺and is denoted by 𝜒(𝐺). If𝐺 is notP-partite for any partitionPinto
finitely many parts, we set𝜒(𝐺):=∞.
Definition 1.1.7(Complete (partite) graphs). We define𝐾(𝑉) := (𝑉 , 𝑉
2
)to be the
complete graph on the vertex-set𝑉 and we write𝐾𝑛:=𝐾( [𝑛]). Given a collection
disjoint setsP ={𝑉1, . . . , 𝑉𝑡}, we denote by𝐾(P) =𝐾(𝑉1, . . . , 𝑉𝑡) the complete𝑡
-partite graph with parts𝑉1, . . . , 𝑉𝑡, i.e. the graph with vertex-set𝑉(𝐺) =𝑉1∪. . .∪𝑉𝑡 and edge-set 𝐸(𝐺) = {𝑒 ∈ 𝑉
2 : |𝑒∩𝑉𝑖| ≤ 1 for all𝑖 ∈ [𝑡]}. Given a graph𝐺 and disjoint sets𝑈1, . . . , 𝑈𝑡, we denote by𝐺[𝑈1, . . . , 𝑈𝑡]the𝑡-partite subgraph of𝐺with
vertex-set𝑈1∪. . .∪𝑈𝑡and edge-set𝐸 = {𝑒 ∈𝐸(𝐺) : |𝑒∩𝑉𝑖| ≤1 for all𝑖 ∈ [𝑡]}.
Definition 1.1.8((Partial) Embeddings). Apartial function 𝑓 : 𝑋 →𝑌is a function 𝑓0: 𝑋0→𝑌 for some 𝑋0⊆ 𝑋. We define therangeof 𝑓 by rg 𝑓 :={𝑦 ∈𝑌 : 𝑓(𝑥)= 𝑦for some𝑥 ∈ 𝑋} and the domain of 𝑓, denoted by dom 𝑓, as the set of all𝑥 ∈ 𝑋
for which 𝑓(𝑥) is defined. A (partial) embeddingof a graph𝐻 into another graph 𝐺 is a (partial) injective function 𝑓 : 𝑉(𝐺) → 𝑉(𝐻) such that 𝑓(𝑢)𝑓(𝑣) ∈ 𝐸(𝐻)
whenever𝑢𝑣 ∈𝐸(𝐺).
Definition 1.1.9(Orderings). Alinear orderon a set𝑋 is a binary relation≤on 𝑋
such that, for all𝑥 , 𝑦 ∈ 𝑋, we have
(i) 𝑥 ≤ 𝑦or𝑦 ≤ 𝑥,
(iii) if𝑥 ≤ 𝑦and𝑦 ≤ 𝑧, then𝑥 ≤ 𝑧.
Observe that every linear order on a finite set 𝑋 induces an enumeration 𝑋 = {𝑥1, 𝑥2, . . .}so that𝑥1 < 𝑥2 < . . ., where we write𝑥 < 𝑦if𝑥 ≤ 𝑦and𝑥≠ 𝑦.
We will now define the graphs which will appear frequently in this thesis. Definition 1.1.10(Paths). For𝑛 ≥ 0, we define thepath𝑃𝑛as the graph with𝑛+1
vertices𝑣1, . . . , 𝑣𝑛+1and𝑛edges𝑣1𝑣2, 𝑣2𝑣3, . . . , 𝑣𝑛𝑣𝑛+1.
Definition 1.1.11(Cycles). For𝑛 ≥3, we define thecycle𝐶𝑛as the graph obtained from𝑃𝑛−1by adding the edge𝑣𝑛𝑣1. For technical reasons, we shall consider a single
vertex and a single edge as degenerate cycles𝐶1and𝐶2.
Definition 1.1.12(Trees). A graph𝑇 is called atreeif it does not contain any finite
cycles of length at least 3 as subgraphs. Arooted treeis a tuple (𝑇 , 𝑟) where𝑇 is
a tree and𝑟 ∈𝑉(𝑇). Given a rooted tree (𝑇 , 𝑟) and 𝑠, 𝑡 ∈𝑉(𝑇), we say that 𝑡 is a
child of𝑡 (and𝑠is theparent of𝑡) if𝑠𝑡 ∈ 𝐸(𝑇) and𝑠 is on the unique path from𝑡
to𝑟. Observe that every vertex other than the root has a unique parent but possibly
many or no children.
Definition 1.1.13(Powers of graphs). Thedistanceof two vertices𝑢, 𝑣 in a graph 𝐻, is the length of the shortest path from 𝑢 to 𝑣 in 𝐻. If no such path exists, the
distance of𝑢and𝑣 is infinite. The 𝑘-th (distance) powerof a graph𝐻, denoted by 𝐻𝑘, is the graph obtained from 𝐻 by adding an edge between any two vertices of
distance at most 𝑘 in𝐻. Popular examples of this are powers of paths and powers
of cycles.
1.1.4 Hypergraphs
Given an integer 𝑘 ≥ 2, a𝑘-uniform hypergraph(for short 𝑘-graph)𝐺 = (𝑉 , 𝐸) is
a tuple consisting of a set𝑉, called thevertices of𝐺, and a set𝐸 ⊆ 𝑉
𝑘
, called the edgesof𝐺. We write𝑉(𝐺) to refer to the vertex-set of𝐺 and𝐸(𝐺) to refer to the
edge-set of𝐺. We further denote the sizes of𝑉(𝐺) and𝐸(𝐺) by 𝑣(𝐺) := |𝑉(𝐺) |
Definition 1.1.14(Degree). Given some 𝑒 ⊆ 𝑉(𝐺) with |𝑒| ∈ [𝑘 −1], thedegree
of 𝑒 is given by deg𝐺(𝑒) := |𝑓 ∈𝐸(𝐻) : 𝑒 ⊆ 𝑓|. If 𝑒 = {𝑣} for some 𝑣 ∈ 𝑉(𝐻)
we simply write deg𝐺(𝑣) for deg𝐺({𝑣}) and if |𝑒| = 𝑘 − 1 we also call deg𝐺(𝑒)
co-degree. If𝐺 is clear from context, we drop the subscript.
Definition 1.1.15(Subgraphs). A 𝑘-graph 𝐻 is called asub-𝑘-graphof a 𝑘-graph 𝐺 if𝑉(𝐻) ⊆ 𝑉(𝐺) and 𝐸(𝐻) ⊆ 𝐸(𝐺). We call𝐻 spanningif𝑉(𝐻) =𝑉(𝐺). If 𝑈 ⊆𝑉(𝐺), we define𝐺[𝑈]to be the subgraph of𝐺induced by𝑈, i.e.𝑉(𝐺[𝑈]) =𝑈
and𝐸(𝐺[𝑈]) ={𝑒 ∈𝐸(𝐺) :𝑒 ⊆𝑈}.
Definition 1.1.16(Independent sets). A set𝐼 ⊆𝑉(𝐺)is calledindependentif there
is no edge𝑒 ∈𝐸(𝐺)with𝑒 ⊆ 𝐼. Theindependence numberof𝐺, denoted by𝛼(𝐺),
is the size of the largest independent set in𝐺. Given a𝑘-graph𝐺 and a partition P = {𝑉1, . . . , 𝑉𝑡} of 𝑉(𝐺), we say that 𝐺 is P-partite if |𝑒 ∩𝑉𝑖| ≤ 1 for every 𝑒 ∈𝐸(𝐻)and every𝑖 ∈ [𝑡]. 𝐺is called𝑡-partite if it isP-partite for some partition P of𝑉 with𝑡parts.
Definition 1.1.17(Complete (partite) graphs). We define𝐾(𝑘)(𝑉) :=(𝑉 , 𝑉
𝑘
) to be
the complete 𝑘-graph on the vertex-set𝑉 and we write𝐾𝑛(𝑘) := 𝐾(𝑘)( [𝑛]). Given a
collection disjoint setsP ={𝑉1, . . . , 𝑉𝑡}, we denote by𝐾(𝑘)(P) =𝐾(𝑘)(𝑉1, . . . , 𝑉𝑡)
the complete𝑡-partite𝑘-graph with parts𝑉1, . . . , 𝑉𝑡, i.e. the𝑘-graph with vertex-set 𝑉(𝐺) =𝑉1∪. . .∪𝑉𝑡 and edge-set𝐸(𝐺) = {𝑒 ∈ 𝑉
𝑘
: |𝑒∩𝑉𝑖| ≤ 1 for all𝑖 ∈ [𝑡]}.
Given a 𝑘-graph 𝐺 and disjoint sets𝑈1, . . . , 𝑈𝑡, we denote by 𝐺[𝑈1, . . . , 𝑈𝑡] the 𝑡-partite subgraph of𝐺 with vertex-set𝑈1∪. . .∪𝑈𝑡and edge-set𝐸 ={𝑒 ∈𝐸(𝐺) : |𝑒∩𝑉𝑖| ≤ 1 for all𝑖 ∈ [𝑡]}. Given some 2≤ 𝑗 ≤ 𝑘−1 and a 𝑗-graph𝐻, we define 𝐾(𝑘)(𝐻) to be the set of all𝑘-cliques in𝐻(𝑗), seen as a𝑘-graph on𝑉.
Definition 1.1.18(Link graphs and (more) degrees). Given a𝑘-graph𝐺 and some 𝑒 ={𝑣1, . . . , 𝑣ℓ} ⊆𝑉(𝐺) withℓ ∈ [𝑘−1], we define thelink-graph Lk𝐺(𝑒) as the (𝑘−ℓ)-graph with vertex-set𝑉(𝐺)and edge-set{𝑓 ∈ 𝑉(𝐺)
𝑘−ℓ
:𝑒∪𝑓 ∈𝐸(𝐺)}.Note
that deg𝐺(𝑒) = |Lk𝐺(𝑒) |. If in addition we are given disjoint sets𝑉1, . . . , 𝑉𝑘−ℓ ⊆ 𝑉(𝐺) \ 𝑒, we denote by Lk𝐺(𝑣1, . . . , 𝑣ℓ;𝑉1, . . . , 𝑉𝑘−ℓ) the (𝑘 − ℓ)-partite (𝑘 − ℓ)-graph with parts 𝑉1, . . . , 𝑉𝑘−ℓ and edge-set {𝑒 ∈ 𝐾
(𝑘−ℓ)
(𝑉1, . . . , 𝑉𝑘−ℓ) : 𝑒 ∪
{𝑣1, . . . , 𝑣ℓ} ∈ 𝐸(𝐺)}. Furthermore, we define deg𝐺(𝑣1, . . . , 𝑣ℓ;𝑉1, . . . , 𝑉𝑘−ℓ) :=
Definition 1.1.19 (Loose path). Theloose 𝑘-uniform path of length𝑚 ≥ 1 is the 𝑘-graph consisting of 𝑚(𝑘−1) +1 distinct linearly ordered vertices and𝑚 edges,
each of which is formed of𝑘consecutive vertices so that consecutive edges intersect
in exactly one vertex. More precisely, its vertex-set is {𝑣1, . . . , 𝑣𝑚(𝑘−1)+1} and its
edge-set is {𝑣1, . . . , 𝑣𝑘},{𝑣𝑘, . . . , 𝑣2𝑘−1}, . . . ,{𝑣(𝑚−1) (𝑘−1)+1, . . . , 𝑣𝑚(𝑘−1)+1}. We
consider a single vertex as a loose path of length 0.
Definition 1.1.20(Loose cycle). Theloose 𝑘-uniform cycleof length𝑚 ≥ 3 is the 𝑘-graph consisting of 𝑚(𝑘 −1) distinct cyclically ordered vertices and 𝑚 edges,
each of which formed of 𝑘 consecutive vertices so that consecutive edges intersect
in exactly one vertex. More precisely, its vertex-set is {𝑣1, . . . , 𝑣𝑚(𝑘−1)} and its
edge-set is{𝑣1, . . . , 𝑣𝑘},{𝑣𝑘, . . . , 𝑣2𝑘−1}, . . . ,{𝑣(𝑚−1) (𝑘−1)+1, . . . , 𝑣1}. We consider
a single vertex as a loose cycle of length 1 and a single edge as a loose cycle of length 2.
Definition 1.1.21 (Tight path). The tight 𝑘-uniform path of length 𝑚 ≥ 1 is the 𝑘-graph with𝑚+𝑘−1 distinct linearly ordered vertices in which any𝑘consecutive
vertices form an edge. More precisely, its vertex-set is {𝑣1, . . . , 𝑣𝑚+𝑘−1} and its edges are{𝑣1, . . . , 𝑣𝑘}, {𝑣2, . . . , 𝑣𝑘+1}, . . ., {𝑣𝑚, . . . , 𝑣𝑚+𝑘−1}. For𝑖 ∈ [𝑘 −1], we consider an independent set of size𝑖 a tight path (all of length 0).
Definition 1.1.22 (Tight cycle). Thetight 𝑘-uniform cycleof length 𝑚 ≥ 1 is the 𝑘-graph with 𝑚 distinct cyclically ordered vertices in which any 𝑘 consecutive
vertices form an edge. More precisely, its vertex-set is {𝑣1, . . . , 𝑣𝑚} and its edges
are {𝑣1, . . . , 𝑣𝑘}, {𝑣2, . . . , 𝑣𝑘+1}, . . ., {𝑣𝑚, . . . , 𝑣𝑘−1}. Note that, for 𝑚 ∈ [𝑘 −1],
the tight cycle of length 𝑚 has no edges and the tight cycle of length 𝑘 is a single
edge.
Definition 1.1.23(Powers of paths and cycles). The𝑝-th power of a𝑘-uniform tight
path (cycle) is the 𝑘-graph obtained by replacing every edge of the (𝑘 + 𝑝 −1)
-uniform tight path (cycle) by the complete 𝑘-graph on 𝑘 + 𝑝− 1 vertices (on the
1.1.5 Edge Colourings and Ramsey Numbers
Since a graph is precisely a 2-graph, we will only define the following concepts for hypergraphs. Given a𝑘-graph𝐺 and a positive integer𝑟 ∈ N, an𝑟-edge-colouring
of𝐺 is an assignment of colours from a list of𝑟 colours to each edge of𝐺. More
formally, an𝑟-edge-colouring of𝐺 is a function 𝜒: 𝐸(𝐺) →𝑆, where𝑆is a set of
size𝑟 (usually 𝑆 = {red,blue} if𝑟 =2 and 𝑆 = [𝑟] if𝑟 ≥ 3). An𝑟-edge-coloured 𝑘-graph is a tuple(𝐺 , 𝜒)of a𝑘-graph𝐺and an𝑟-edge-colouring 𝜒of𝐺. We will
frequently abuse notation and write𝑟-edge-coloured𝑘-graph𝐺.
Two 𝑘-graphs𝐻 and𝐺 are isomorphic if there is a bijection 𝑓 :𝑉(𝐻) →𝑉(𝐺)
such that 𝑒 ∈ 𝐸(𝐻) if and only if 𝑓(𝑒) := {𝑓(𝑣) : 𝑣 ∈ 𝑒} ∈ 𝐸(𝐺). Given an 𝑟-edge-coloured 𝑘-graph 𝐺 and a 𝑘-graph𝐻, a monochromatic copy of 𝐻 in𝐺 is
a (not necessarily induced) subgraph 𝐻0of𝐺 such that𝐻0is isomorphic to 𝐻 and
every edge of𝐻0receives the same colour.
Definition 1.1.24. The𝑟-colour𝑘-uniformRamsey numberof a𝑘-graph𝐺, denoted
by R(𝑘)
𝑟 (𝐺), is the smallest integer 𝑁 such that every 𝑟-coloured 𝐾 (𝑘)
𝑁 contains a monochromatic copy of 𝐺. If 𝑘 = 2, we drop the superscript (𝑘) and simply say 𝑟-colour Ramsey number of𝐺. If𝑟 =2, we drop the subscript𝑟 and simply say 𝑘
-uniform Ramsey number of𝐺. If𝑟 =𝑘 =2, we drop both superscript and subscript
and simply say Ramsey number of𝐺.
We will make the following definition only for ordinary graphs (i.e. for𝑘=2).
Definition 1.1.25 (Coloured subgraphs, neighbourhoods and degrees). Given an edge-coloured graph 𝐺 and a colour𝑖, we define 𝐺𝑖 to be the spanning subgraph
of 𝐺 with all edges of colour 𝑖. If 𝐺 is clear from context and we are given
a vertex 𝑣 ∈ 𝑉(𝐺) and a set 𝑈 ⊆ 𝑉(𝐺), we write 𝑁𝑖(𝑣 , 𝑈) := 𝑁𝐺𝑖(𝑣 , 𝑈) and deg𝑖(𝑣 , 𝑈) :=deg𝐺
𝑖(
𝑣 , 𝑈). If𝑈 =𝑉(𝐺), we simply write𝑁𝑖(𝑣) and deg𝑖(𝑣).
1.1.6 Infinite Graph Theory
Definition 1.1.26(Upper and lower density). Theupper densityof a set 𝐴 ⊆ Nis
defined as
d(𝐴)=lim sup
𝑡→∞
|𝐴∩ [𝑡] | 𝑡
and the upper density of a graph𝐺 with𝑉(𝐺) ⊆Nis given by the upper density of
its vertex-set, i.e. d(𝐺) := d(𝑉(𝐺)). Thelower density d(𝐴) (or d(𝐺)) is defined
similarly in terms of the infimum and we speak of thedensity, whenever lower and upper density coincide.
Definition 1.1.27(Ramsey upper density). The𝑟-colour Ramsey upper densityof𝐺,
denoted by Rd𝑟(𝐺), is the supremum over all𝑑 ≥ 0 such that every𝑟-edge-coloured
𝐾
N contains a monochromatic copy of𝐺 with d(𝐺) ≥ 𝑑. If 𝑟 = 2, we drop the subscript.
We will now define the infinite graphs which will appear frequently in this thesis. Definition 1.1.28(Infinite path). Theinfinite path 𝑃∞ is the graph with vertex-set 𝑉(𝑃) = {𝑣𝑖 :𝑖 ∈N}and edge-set𝐸(𝑃) ={𝑣𝑖𝑣𝑖+1:𝑖 ∈ N}. 𝑃∞is sometimes called
one-way infinite path. The two-way infinite path 𝐶∞ is the graph with vertex-set 𝑉(𝑃) = {𝑣𝑖 : 𝑖 ∈ Z} and edge-set 𝐸(𝑃) = {𝑣𝑖𝑣𝑖+1 : 𝑖 ∈ Z}. 𝐶∞ can be seen as the
infinite analogue of a cycle.
1.1.7 Random Graphs
A (finite) probability space is a tuple (Ω,P) of a finite set Ω and a probability
function, that is a function P : 2Ω → [0,1]
R with P[∅] = 0, P[Ω] = 1 and P[𝐴∪𝐵] = P[𝐴] +P[𝐵] for all disjoint sets 𝐴, 𝐵 ⊆ Ω. Note that Pis uniquely
determined by the values ofP[𝑥] :=P[{𝑥}]for all𝑥 ∈Ω.
Definition 1.1.29(Conditional Probability). Given a finite probability space(Ω,P)
and events 𝐴, 𝐵 ⊆ Ω with P[𝐵] > 0, we define the conditional probability of 𝐴
given𝐵asP[𝐴|𝐵] := P[
𝐴∩𝐵] P[𝐵] .
Definition 1.1.30 (With high probability). For each 𝑛 ∈ 𝑁, let 𝐸𝑛 be an event in
some probability space(Ω𝑛,P𝑛). We say that the event𝐸 = (𝐸𝑛)𝑛∈Nholdswith high probability (w.h.p.) if lim𝑛→∞P[𝐸𝑛] =1.
Arandom variableis a function 𝑋 :Ω→ 𝑆for some set 𝑆. If 𝑆 ⊆ R, we denote
theexpected valueof𝑋 by
E[𝑋] := Õ 𝑥∈Ω P[𝑥] ·𝑋(𝑥) = Õ 𝑦∈𝑆 P[𝑋 = 𝑦] ·𝑦,
where we denoteP[𝑋 =𝑦] :=P[{𝑥 ∈Ω : 𝑋(𝑥)= 𝑦}]. Thedistributionof a random
variable is the probability function P𝑋 : 2
𝑆 → [0
,1]R, where P𝑋 is induced by
P𝑋(𝑦) := P[𝑋 =𝑦]. On the other hand, given a probability space (𝑆,P), the random variable 𝑋 : 𝑆 → 𝑆 defined by 𝑋(𝑦) = 𝑦 has distribution P. Therefore,
when the original probability space is not important, we often identify a random variable with its distribution. Arandom graph is a random variable taking values in a set of graphs. We will usually only be interested in its distribution. We will be dealing with the following random graphs in this thesis. Here, given a (hyper-)graph
𝐺, we denote byG𝐺 the set of all spanning subgraphs of𝐺and we writeG𝑛 :=G𝐾𝑛 andG𝑛(𝑘) := G
𝐾( 𝑘)
𝑛 , and we denote by
G𝑛,𝑚 the set of all spanning subgraphs of𝐾𝑛 with exactly𝑚 edges.
Definition 1.1.31(Random graphs). Let𝑛 ∈Nand let 𝑝 ∈ [0,1]R.
• 𝐺(𝑛, 𝑝) is the random spanning subgraph of𝐾𝑛in which every edge of𝐾𝑛is
present independently with probability 𝑝. Its distribution P : 2G𝑛 → [0,1]
R is induced byP[𝐺(𝑛, 𝑝) =𝐻] = 𝑝𝑒(𝐻)(1− 𝑝)𝑒(𝐾𝑛)−𝑒(𝐻) for𝐻 ∈ G𝑛.
• Given𝑚 ∈Nwith 0 ≤ 𝑚 ≤ 𝑛
2, we define𝐺(𝑛, 𝑚)to be the random spanning
subgraph of 𝐾𝑛 which is chosen uniformly from all spanning subgraphs of 𝐾𝑛 with exactly 𝑚 edges. Its distributionP : 2G𝑛, 𝑚 → [0,1]
R is induced by P[𝐺(𝑛, 𝑚) =𝐻] =1/ 𝑒(𝐾𝑛)
𝑚
for𝐻 ∈ G𝑛,𝑚.
• Given 𝑘 ∈ N, we define𝐺(𝑘)(𝑛, 𝑝) to be the random spanning subgraph of 𝐾(
𝑘)
𝑛 in which every edge of 𝐾 (𝑘)
𝑛 is present independently with probability
𝑝. Its distribution P : 2G (𝑘) 𝑛 → [0 ,1] R is induced by P 𝐺(𝑘)(𝑛, 𝑝)= 𝐻 = 𝑝𝑒(𝐻)(1−𝑝)𝑒(𝐾 (𝑘) 𝑛 )−𝑒(𝐻) for 𝐻 ∈ G𝑛(𝑘).
• Given a graph 𝐺, we define 𝐺𝑝 to be the random spanning subgraph of
𝐺 in which every edge of 𝐺 is present independently with probability 𝑝.
Its distribution P : 2G𝐺 → [0,1]
R is induced by P
𝐺𝑝 =𝐻 = 𝑝𝑒(𝐻)(1− 𝑝)𝑒(𝐺)−𝑒(𝐻) for𝐻 ∈ G𝐺.
Definition 1.1.32(Graph properties and threshold probabilities). Agraph property
be a graph property whose members are subgraphs of𝐾𝑛 for some𝑛 ∈ N and let 𝑝0:N→ [0,1]
Rbe a function.
• 𝑝0is called acoarse threshold probabilityforPif
lim 𝑛→∞P[ 𝐺(𝑛, 𝑝(𝑛)) ∈ P] = 0 if𝑝 =𝑜(𝑝0), 1 if𝑝 =𝜔(𝑝0).
• 𝑝0is called asharp threshold probabilityforPif for every𝜀 >0, we have
lim 𝑛→∞P[ 𝐺(𝑛, 𝑝(𝑛)) ∈ P] = 0 if𝑝(𝑛) ≤ (1−𝜀)𝑝0(𝑛)for all𝑛 ∈N, 1 if𝑝(𝑛) ≥ (1+𝜀)𝑝0(𝑛) for all𝑛∈N.
Note that threshold probabilities do not necessarily exist for all graph properties. However, Bollobás and Thomason [12] proved that coarse thresholds exist for all monotonegraph properties (Pis monotone if𝐻 ∈ P and𝐻 ⊆ 𝐺 implies𝐺 ∈ P).
1.2 Graphs with Linear Ramsey Number
Investigating graphs with linear Ramsey number has been one of the most studied topics in Ramsey Theory, motivated particularly by a series of conjectures of Burr and Erdős [16, 17]. A lot of work presented in this thesis was inspired by results from this area, and hence we will start with a brief introduction to it.
A sequence of graphs F = {𝐹1, 𝐹2, . . .}3is said to havelinear𝑟-colour Ramsey
numberif we have R𝑟(𝐹𝑛) =𝑂(𝑛). If𝑟 =2, we will simply sayF has linear Ramsey number. The following observation shows that graphs with linear Ramsey number must be quite “sparse”.
Observation 1.2.1. IfF = {𝐹1, 𝐹2, . . .}is a sequence of graphs with linear Ramsey
number, then we have𝑒(𝐹𝑛) =𝑂(𝑛log(𝑛)).
3Recall thatF = {𝐹1, 𝐹2, . . .}is a sequence of graphs if 𝐹𝑛 is a graph with𝑛 vertices for every
Proof. SinceF has linear Ramsey number, there is some constant𝐶 ∈Nsuch that
R(𝐹𝑛) ≤𝐶 𝑛for all𝑛 ∈N. We will show that𝑑(𝐹𝑛) ≤2 log2(𝐶 𝑛) +2/𝑛. Fix some 𝑛∈Nand let𝑁 :=𝐶 𝑛. Colour the edges of𝐾𝑁uniformly at random with 2 colours
and let𝐸𝑛be the event that there is a monochromatic copy of𝐹𝑛. Since𝑁 ≥ R(𝐹𝑛),
we have P[𝐸𝑛] = 1. On the other hand we have P[𝐸𝑛] ≤ 2· 𝑁𝑛 ·2−𝑒(𝐹𝑛). We
conclude that𝑒(𝐹𝑛) ≤log2(2𝑁𝑛) =1+𝑛log2(𝐶 𝑛), concluding the proof.
1.2.1 Paths and Cycles
Paths and cycles are two of the most elementary examples of graphs with linear Ramsey number. While it is not hard to see that paths have linear Ramsey number, there has been a lot of work in trying to determine their Ramsey numbers exactly. For two colours, Gerencsér and Gyárfás [55] proved the following result.
Theorem 1.2.2(Gerencsér–Gyárfás [55]). We haveR(𝑃𝑛−1) =b3𝑛/2−1cfor every 𝑛 ≥ 2.
Confirming a conjecture of Faudree and Schelp [50] for large𝑛, Gyárfás, Ruszinkó,
Sárközy and Szemerédi [65] proved that R3(𝑃𝑛−1) = 2𝑛−2 if𝑛is even 2𝑛−1 if𝑛is odd.
for every large enough𝑛 ∈N. For more than three colours, less is known. It easily
follows from a result of Erdős and Gallai [38] that R𝑟(𝑃𝑛−1) ≤ 𝑟 𝑛 and Bierbrauer and Gyárfás [10] proved that R𝑟(𝑃𝑛−1) ≥ (𝑟−1−𝑜(1))𝑛. Recently, there has been a lot of work on this problem and, after progress by Sárközy [109], and Davies, Jenssen and Roberts [32], the currently best known upper bound is due to Knierim and Su [79], who proved that R𝑟(𝑃𝑛−1) ≤ (𝑟−1/2+𝑜(1))𝑛.
Clearly we have R𝑟(𝐶𝑛) ≥R𝑟(𝑃𝑛−1)for all𝑛, 𝑟 ≥ 2 since𝐶𝑛contains𝑃𝑛−1and it turns out the two numbers are very similar for even cycles. Faudree and Schelp [49] and independently Rosta [103] showed that R(𝐶𝑛) = 3𝑛/2−1 for all even 𝑛 ≥ 6
and Benevides and Skokan [8] showed that R3(𝐶𝑛) = 2𝑛 for all sufficiently large
R𝑟(𝐶𝑛) ≤ (𝑟 +𝑜(1))𝑛 and all recent result about paths above ([32, 79,109]) came with extensions to even cycles, in particular we have R𝑟(𝐶𝑛) ≤ (𝑟−1/2+𝑜(1))𝑛.
Interestingly, the problem is very different for odd cycles. Bondy and Erdős [13] showed that R(𝐶𝑛)=2𝑛−1 for all odd𝑛 ≥ 5 and, confirming a conjecture of Bondy
and Erdős, Jenssen and Skokan [74] showed that R𝑟(𝐶𝑛) = 2 𝑟−1(
𝑛−1) +1 for all
sufficiently large odd𝑛.
1.2.2 Trees
Trees are another well-studied family of graphs in Ramsey theory. It is again easy to show that trees have linear𝑟-colour Ramsey number. Indeed, let us prove that
R𝑟(𝑇) ≤2𝑟(𝑛−1) +1 for every tree𝑇with𝑛vertices. Let𝑁 =2𝑟(𝑛−1) +1 and let
𝐾𝑁 be edge-coloured with𝑟 colours. By the pigeonhole principle, one colour (say blue) appears at least 𝑁
2/𝑟times and hence the average degree of the blue subgraph
is at least (𝑁 −1)/𝑟. Therefore, there is a blue subgraph with minimum degree at
least(𝑁−1)/(2𝑟) ≥𝑛−1.4 It is now easy to greedily embed𝑇 into this subgraph.
A famous conjecture of Burr and Erdős [17], which was solved for sufficiently large𝑛 by Zhao [117], states that this can be improved roughly by a factor of 2 in
the case of two colours.
Conjecture 1.2.3(Burr–Erdős [17]). For every tree𝑇on𝑛vertices, we haveR(𝑇) ≤
2𝑛−2if𝑛is even andR(𝑇) ≤ 2𝑛−3if𝑛is odd.
1.2.3 Graphs with Bounded Degree
Burr and Erdős also studied Ramsey numbers of graphs with bounded degree and made the following conjecture, which states that every sequence of graphs with bounded degree has linear Ramsey number.
Conjecture 1.2.4(Burr–Erdős [16]). For allΔ≥ 1, there exists some𝑐 =𝑐(Δ) > 0
such thatR(𝐺) ≤ 𝑐𝑛for every graph𝐺on𝑛vertices with maximum degree at most
Δ.
4It is a well-known combinatorial fact that every graph of average degree 𝑑 has a subgraph of minimum degree at least𝑑/2.
Conjecture1.2.4was solved by Chvatál, Rödl, Szemerédi, Trotter [21] in an early application of the regularity lemma. Since then, there has been many improvements to the constant 𝑐(Δ). After an improvement of Eaton [36], Graham, Rödl and
Ruciński [57] came close to settling the problem by showing that the constant can be chosen to be𝑐(Δ) =2𝑂(Δlog2(Δ)) and providing the following lower bound.
Theorem 1.2.5 (Graham–Rödl–Ruciński [57]). There is a sequence of bipartite graphsF ={𝐹1, 𝐹2, . . .}withR(𝐹𝑛) =2
Ω(Δ)
𝑛.
The currently best known upper bound is due to Conlon, Fox and Sudakov [26]. Theorem 1.2.6(Conlon–Fox–Sudakov [26]). There is some constant 𝑐 > 0 such
thatR(𝐺) ≤ 2𝑐Δlog(Δ)𝑛 for every graph 𝐺 on𝑛 vertices with maximum degree at
mostΔ.
Furthermore, Conlon [23], and independently Fox and Sudakov [52] were able to remove the extra log(Δ)-factor in the exponent for bipartite graphs with maximum
degree at most Δ. Allen, Brightwell and Skokan [5] showed that the constant in
Theorem 1.2.6 can be significantly improved to for a wide range of graphs. The bandwidth of a graph 𝐺, denoted by bw(𝐺), is the smallest integer 𝑘 for which
there is a bijection 𝑓 :𝑉(𝐺) → [𝑣(𝐺)]such that|𝑓(𝑢) − 𝑓(𝑣) | ≤ 𝑘 for every edge 𝑢𝑣 ∈𝐸(𝐺).
Theorem 1.2.7 (Allen–Brightwell–Skokan [5]). For every Δ ∈ N, there are con-stants 𝛽 > 0and𝑛0 ∈ N such thatR(𝐺) ≤ (2𝜒(𝐺) +4)𝑛 for every graph𝐺 with 𝑛 ≥ 𝑛0vertices, maximum degree at mostΔand bandwidth at most𝛽𝑛.
Finally, let us mention another famous conjecture of Burr and Erdős regarding graphs with bounded degeneracy.
Conjecture 1.2.8(Burr–Erdős [16]). For all𝑑 ≥ 1, there exists some𝐶 =𝐶(𝑑) > 0
such thatR(𝐺) ≤ 𝐶 𝑛for every𝑑-degenerate graph𝐺 on𝑛vertices.
1.3 Monochromatic Graph Tiling Problems
1.3.1 History
The first result in this area is due to Gerencsér and Gyárfás [55]. Recall that single vertices and edges are considered (degenerate) paths and cycles in this thesis. Theorem 1.3.1 (Gerencsér–Gyárfás [55]). The vertices of every 2-edge-coloured complete graph on𝑛vertices can be partitioned into two monochromatic paths, one
of each colour.
A conjecture of Lehel (which first appeared in the PhD thesis of Ayel [7] in 1979, where it was also proved for some special types of colourings of𝐾𝑛) states that the
same should be true for cycles instead of paths. Almost 20 years later, in 1998, Łuczak, Rödl and Szemerédi [92] proved Lehel’s conjecture for all sufficiently large
𝑛using the regularity method. In [2], Allen gave an alternative proof, which gave
a better bound on𝑛. Finally, Bessy and Thomassé [9] proved Lehel’s conjecture for
all integers𝑛 ≥ 1.
Theorem 1.3.2(Bessy–Thomassé [9]). The vertices of every2-edge-coloured com-plete graph on𝑛vertices can be partitioned into two monochromatic cycles, one of
each colour.
The problem was soon extended to multiple colours.
Theorem 1.3.3 (Gyárfás [59]). The vertices of every 𝑟-edge-coloured complete
graph on𝑛vertices can be covered by𝑂(𝑟4)monochromatic paths.
Theorem 1.3.4(Erdős–Gyárfás–Pyber [42]). The vertices of every𝑟-edge-coloured
complete graph on 𝑛 vertices can be partitioned into 𝑂(𝑟2log𝑟) monochromatic
cycles.
It is a major open problem to determine cp(𝑟), the smallest number of cycles
needed in Theorem1.3.4. More precisely, cp(𝑟) is the smallest integer𝑡 such that
the vertices of every 𝑟-edge-coloured 𝐾𝑛 can be partitioned into a collection of at most 𝑡 vertex-disjoint monochromatic cycles. Note that (without Theorem 1.3.4)
it is not clear at all whether cp(𝑟) is finite (i.e. independent of 𝑛, the size of the
host-graph).
Problem 1.3.5. What iscp(𝑟)?
It was further conjectured in [42] that cp(𝑟) =𝑟 for all𝑟 ∈ N. This conjecture
was refuted however by Pokrovskiy [95], who showed that cp(𝑟) > 𝑟 for every 𝑟 ≥ 3. Gyárfás, Ruszinkó, Sárközy and Szemerédi [62] showed that 𝑂(𝑟log𝑟)
cycles suffice for all sufficiently large𝑛.
Pokrovskiy proposed the following slightly weaker conjecture.
Conjecture 1.3.6 (Pokrovskiy [95]). For every positive integer 𝑟, there is some
constant𝑐 =𝑐(𝑟), such that in every𝑟-edge-coloured𝐾𝑛, there is a collection of𝑟
vertex-disjoint monochromatic cycles covering all but at most𝑐vertices.
It is still possible that𝑟 monochromatic paths suffice.
Conjecture 1.3.7 (Gyárfás [59]). The vertices of every𝑟-edge-coloured complete
graph on𝑛vertices can be partitioned into𝑟monochromatic paths.
1.3.2 New Results
Hypergraph Cycles
It is natural to ask if cycle partition problems like Theorem1.3.4can be generalised to hypergraphs. Such questions were first studied by Gyárfás and Sárközy [64] who proved the following result about loose cycles.
Theorem 1.3.8(Gyárfás–Sárközy). For every 𝑘, 𝑟 ∈ N, there is some 𝑐 = 𝑐(𝑘 , 𝑟)
such that the vertices of every𝑟-edge-coloured complete𝑘-graph can be partitioned
into a collection of at most𝑐loose cycles.
Later, Sárközy [107] showed that 𝑐(𝑘 , 𝑟) can be be chosen to be 50𝑟 𝑘log(𝑟 𝑘).
Gyárfás [60] conjectured that a similar result can be obtained for tight cycles (again, a single vertex is considered a tight cycle here).
Conjecture 1.3.9(Gyárfás [60]). For every 𝑘 , 𝑟 ∈ N, there is some 𝑐 = 𝑐(𝑘 , 𝑟) so
that the vertices of every𝑟-edge-coloured complete𝑘-graph can be partitioned into
We prove this conjecture and a generalisation in which we allow the host-hypergraph to be any𝑘-graph with bounded independence number (the main
moti-vation of this generalisation is an interesting application discussed below).
Theorem 1.3.10 (Bustamante–Corsten–Frankl–Pokrovskiy–Skokan [18]). For ev-ery 𝑘 , 𝑟 , 𝛼 ∈ N, there is some𝑐 = 𝑐(𝑘 , 𝑟 , 𝛼) such that the vertices of every𝑟
-edge-coloured 𝑘-graph𝐺 with independence number 𝛼(𝐺) ≤ 𝛼 can be partitioned into
at most𝑐monochromatic tight cycles.
The proof of Theorem1.3.10, which is based on the absorption method and the hypergraph regularity method, will be presented in Section3.1.
Tiling Number of Sequences of Graphs
We turn our attention back to graphs now considering other “tiles” than cycles. We investigate problems in which we are given a sequence of graphs F and an
edge-coloured complete graph𝐾𝑛and our goal is to partition𝑉(𝐾𝑛) into few
monochro-matic copies of graphs from F. An F-tiling T of a graph 𝐺 is a collection of
vertex-disjoint copies of graphs from F in 𝐺 with𝑉(𝐺) = Ð
𝑇∈T𝑉(𝑇). If 𝐺 is edge-coloured, we say thatT ismonochromaticif every𝑇 ∈ T is monochromatic
(not necessarily in the same colour). Using this notation, Theorem1.3.4states that in every𝑟-edge coloured𝐾𝑛, we can find a monochromaticC-tiling of size at most 𝑂(𝑟2log𝑟), whereCis the family of all cycles (including a single vertex and a single
edge).
Definition 1.3.11(Tiling number). Let𝜏𝑟(F, 𝑛) be the minimum𝑡 ∈Nsuch that in
every𝑟-edge-coloured𝐾𝑛there is a monochromatic F-tiling of size at most𝑡. We
call𝜏𝑟(F ) =sup𝑛∈
N𝜏𝑟(F, 𝑛) the𝑟-colourtiling numberof F. If𝑟 =2, we simply write𝜏(F, 𝑛)and𝜏(F ), and simply saytiling number.
The results of Pokrovskiy and of Erdős, Gyárfás and Pyber above imply that, for all𝑟 ≥ 3,
𝑟+1≤ 𝜏𝑟(C) ≤𝑂(𝑟2log𝑟).
Note that, in general, it is not clear at all that𝜏𝑟(F ) is finite and it is a natural question to ask for which families this is the case. The study of such tiling problems
for more general families of graphs was initiated by Grinshpun and Sárközy [58] who proved the following result. LetFΔbe the collection of all sequences of graphs
F withΔ(F ) ≤Δ.
Theorem 1.3.12 (Grinshpun–Sárközy [58]). We have 𝜏2(F ) ≤ 2𝑂(ΔlogΔ) for all F ∈ FΔ. In particular,𝜏2(F )is finite wheneverΔ(F )is finite.
Grinshpun and Sárközy also proved that 𝜏2(F ) ≤ 2𝑂(Δ) for every sequence of
bipartite graphsF of maximum degree Δand showed that this is best possible up
to the implicit constant (see Theorem1.3.19). Sárközy [108] further proved that the constant in Theorem1.3.12can be improved a lot for the special case of powers of cycles.
For more than two colours not much is known. Elekes, D. Soukup, L. Soukup and Szentmiklóssy [37, Problem 6.4] asked the following problem after proving a similar statement for infinite graphs.5
Problem 1.3.13 (Elekes et al. [37]). Prove that for every𝑟 , 𝑝 ∈ N, there is some 𝑐 = 𝑐(𝑟 , 𝑝) such that the vertices of every𝑟-edge-coloured complete graph can be
partitioned into at most𝑐monochromatic 𝑝-th powers of cycles.
We answer this problem positively. In fact, we obtain the following generalisation to hypergraphs and host-graphs with bounded independence number as a corollary of Theorem1.3.10.
Theorem 1.3.14 (Bustamante–Corsten–Frankl–Pokrovskiy–Skokan [18]). For ev-ery 𝑘 , 𝑟 , 𝑝, 𝛼 ∈ N, there is some 𝑐 = 𝑐(𝑘 , 𝑟 , 𝑝, 𝛼) such that the vertices of every 𝑟-edge-coloured 𝑘-graph 𝐺 with 𝛼(𝐺) ≤ 𝛼 can be partitioned into at most 𝑐
monochromatic𝑝-th powers of tight cycles.
Grinshpun and Sárközy [58] conjectured that their Theorem1.3.12should extend to𝑟 colours as well.
Conjecture 1.3.15(Grinshpun–Sárközy [58]). For every𝑟 ∈N, there is some𝑐𝑟 > 0
so that𝜏𝑟(F ) ≤2Δ
𝑐𝑟
for allΔ≥ 1and all F ∈FΔ.
5The problem is phrased differently in [37] but this version is stronger, as Elekes et al. explain below
Note that it is still open to decide whether𝜏𝑟(F ) is finite for such families. We
show that this is the case and make progress towards Conjecture1.3.15. Recall that exp𝑘 denotes the
𝑘-th composition of the exponential function.
Theorem 1.3.16(Corsten–Mendonça). There is an absolute constant 𝐾 > 0 such
that for all integers𝑟 ,Δ≥ 2and all F ∈ FΔ, we have𝜏𝑟(F ) ≤ exp3 𝐾 𝑟2Δ3. In
particular,𝜏𝑟(F ) < ∞wheneverΔ(F ) < ∞.
Theorem1.3.16also provides a positive answer to Problem1.3.13and the bound is much better than the implicit bound in Theorem 1.3.14. The proof of Theo-rem1.3.16, which is based on the absorption method and the regularity method, will be presented in Section3.2.
Lower Bounds and Graphs with Linear Ramsey Number
Sequences of graphs with finite tiling number are closely related to sequences with linear Ramsey number. Observe that a sequence of graphs F = {𝐹1, 𝐹2, . . .} has
linear 𝑟-colour Ramsey number if and only if there exists some 𝜌 > 0 such that
every𝑟-edge-coloured𝐾𝑛contains a monochromatic copy of𝐹𝑚 for every𝑚 ≤ 𝜌𝑛. A slightly weaker condition (in which we require only one monochromatic copy of 𝐹𝑚 for some 𝑚 ≥ 𝛿𝑛) is a necessary condition to have finite tiling number:
given a sequence of graphsF, it follows from the pigeonhole principle that every 𝑟-edge-coloured𝐾𝑛contains a monochromatic copy fromF of size at least𝑛/𝜏𝑟(F ).
Define𝜌𝑟(F )to be the supremum over all𝜌 ≥ 0 such that every𝑟-edge coloured𝐾𝑛
contains a monochromatic copy from F of size at least𝜌𝑛. The above application
of the pigeonhole principle gives the following observation.
Observation 1.3.17. We have𝜏𝑟(F ) ≥1/𝜌𝑟(F ) for every sequence of graphsF.
Note that𝜌𝑟(F ) >0 for every familyF with linear𝑟-colour Ramsey number but
the converse is not always true.6 It is however true if F is an increasing family of
graphs (i.e. if𝐹𝑛 ⊆ 𝐹𝑛+1for every𝑛∈N), in which case we have 𝜌𝑟(F ) = inf
𝑛∈N(
𝑛/R𝑟(𝐹𝑛)) (1.3.1)
6For example, consider the sequenceF ={𝐹1, 𝐹2, . . .}where𝐹𝑛is an independent set for all even
and thus, using Observation1.3.17,
𝜏𝑟(F ) ≥sup 𝑛∈N
(R𝑟(𝐹𝑛)/𝑛). (1.3.2)
This relationship can be used to translate existing lower bounds for Ramsey numbers of sequences of graphs to lower bounds for their tiling number. While (1.3.2) can only be directly applied to increasing sequences of graphs, Grinshpun and Sárközy [58] observed that one can modify sequences of bounded degree to avoid this problem. Observation 1.3.18. For every sequence of graphsF ={𝐹1, 𝐹2, . . .}with maximum
degree at mostΔandR(𝐹𝑛) ≥𝐶 𝑛for every𝑛∈N, there is an increasing sequence
of graphsF˜ = {𝐹˜1,𝐹˜2, . . .} with maximum degreeΔandR(𝐹˜𝑛) ≥ 𝐶 𝑛/4for every 𝑛∈N.
Combining this with Theorem1.2.5 of Graham, Rödl and Ruciński and (1.3.2) immediately gives the following lower bound, which almost matches the upper bound in Theorem1.3.12.
Theorem 1.3.19(Grinshpun–Sárközy [58]). There is a sequence of graphsF ∈FΔ
with𝜏2(F ) ≥2Ω(Δ).
Observation 1.3.17 further implies that 𝜌𝑟(F ) > 0 (or having linear 𝑟-colour
Ramsey number if F is increasing) is a necessary condition for F to have finite
tiling number. In the other direction, we have 𝜏𝑟(F, 𝑛) ≤ log(𝑛)/𝜌𝑟(F ) (this
follows by greedily taking the largest monochromatic copy from F among the
uncovered vertices until all vertices are covered). The following example, which was provided by Alexey Pokrovskiy (personal communication), shows that this is essentially tight and therefore the above necessary condition is not sufficient. Let𝑆𝑛 be a star with𝑛 vertices and letS = {𝑆1, 𝑆2, . . .} be the family of stars. It follows
readily from the pigeonhole principle that 𝑅𝑟(𝑆𝑛) ≤ 𝑟(𝑛−2) +2 for every 𝑛 ∈ N
and thus 𝜌𝑟(S) ≥ 1/𝑟. However,𝜏𝑟(S)=∞for every𝑟 ≥ 2.
Example 1.3.20. For every𝑟 ≥ 2, we have𝜏𝑟(S, 𝑛) ≥𝑟·log𝑛/8 for all sufficiently
Proof. Let 𝜏 = 𝑟log𝑛/8 and colour 𝐸(𝐾𝑛) uniformly at random with 𝑟 colours.
Given a vertex𝑣 ∈ [𝑛] and a colour𝑐, let𝑆𝑐(𝑣) be the star centred at𝑣 formed by
all the edges of colour𝑐incident on𝑣. Note that there is a monochromaticS-tiling
of size at most 𝜏 if and only if there are distinct vertices 𝑣1, . . . , 𝑣𝜏 and colours 𝑐1, . . . , 𝑐𝜏 ∈ [𝑟]such thatÐ
𝑖∈[𝜏]𝑉(𝑆𝑐𝑖(𝑣𝑖)) =[𝑛].
Fix distinct vertices 𝑣1, . . . , 𝑣𝜏 ∈ [𝑛] and colours 𝑐1, . . . , 𝑐𝜏 ∈ [𝑟]. Let 𝑈 be
the random set 𝑈 = Ð𝑖∈[𝜏]𝑉(𝑆𝑐𝑖(𝑣𝑖)). Notice that the events {𝑣 ∈ 𝑈}, 𝑣 ∈ [𝑛] \{𝑣1, . . . , 𝑣𝜏}, are independent and each has probability 1− (1−1/𝑟)
𝜏. Therefore, using𝑒−𝑥/(1−𝑥) ≤ 1−𝑥 ≤ 𝑒𝑥for all𝑥 ≤ 1, we get
P[𝑈 = [𝑛]] = (1− (1−1/𝑟)𝜏)𝑛−𝜏 ≤ exp(−(𝑛−𝜏) (1−1/𝑟)𝜏) ≤ exp−𝑛(1−1/𝑟)𝜏+1
≤ exp(−𝑛exp(−4𝜏/𝑟)) ≤ exp −√𝑛.
Taking a union bound over all choices of𝑣1, . . . , 𝑣𝜏 and𝑐1, . . . , 𝑐𝜏, we conclude that
the probability that there is a monochromaticS-tiling of size𝜏is at most (𝑟 𝑛)−𝜏 ·𝑒−
√ 𝑛
<1
for all sufficiently large 𝑛. Hence, there exists an𝑟-colouring of 𝐸(𝐾𝑛) without a monochromaticS-tiling of size at most𝜏, finishing the proof.
Recall that Lee [90] proved that sequences of graphs with bounded degeneracy have linear Ramsey number. Since every star has degeneracy 1, Example 1.3.20
further shows that it is not possible to extend this to a tiling result. Nevertheless, it may be possible to allow unbounded degrees in this case.
Question 1.3.21. Is there a function𝜔 :N→ ∞withlim𝑛→∞𝜔(𝑛) =∞, such that
the following is true for all integers𝑟 , 𝑑 ≥ 2? If F = {𝐹1, 𝐹2, . . .} is a sequence
of 𝑑-degenerate graphs with 𝑣(𝐹𝑛) = 𝑛 and Δ(𝐹𝑛) ≤ 𝜔(𝑛) for all 𝑛 ∈ N, then 𝜏𝑟(F ) < ∞.
It is possible to obtain such a result for graphs of bounded arrangeability (a concept in between bounded degree and bounded degeneracy) as explained in Section3.2.5.
1.4 Ramsey Problems for Infinite Graphs
1.4.1 Infinite Paths
Theorem1.2.2implies that that every 2-edge-coloured𝐾𝑛contains a monochromatic path withb(2𝑛+1)/3cvertices. In Section4.1we will investigate a similar problem
concerning the infinite path𝑃∞, where, given a 2-edge-coloured𝐾N, we try to find a
monochromatic infinite path with the largest possible upper density. Since this path does not need to respect the order ofN, we cannot easily infer any bounds from the finite case. Due to the following result of Erdős and Galvin [39], we will restrict our attention to upper densities.
Theorem 1.4.1 (Erdős–Galvin [39]). There is a2-edge-colouring of 𝐾
N in which every monochromatic infinite path has lower density0.
Recall the definition of Ramsey upper density, Definition 1.1.27. Probably the first result in this direction is due to Rado [99].
Theorem 1.4.2(Rado [99]). Every𝑟-edge-coloured 𝐾
N contains a collection of at most𝑟 vertex-disjoint monochromatic paths covering all vertices. In particular, one
of them has upper density at least1/𝑟.
Interestingly, this gives a positive answer of an infinite analogue of Conjec-ture1.3.7. For two colours, this further implies that we can always find a monochro-matic path of upper density at least 1/2. Erdős and Galvin [39] improved this special
case.
Theorem 1.4.3(Erdős–Galvin [39]). We have2/3≤ Rd(𝑃∞) ≤ 8/9.
DeBiasio and McKenney [33] recently improved the lower bound to 3/4 and
conjectured the correct value to be 8/9. Progress towards this conjecture was
made by Lo, Sanhueza-Matamala and Wang [91], who raised the lower bound to (9+√17)/16 ≈ 0.82019. We settle the problem of determining the Ramsey
upper density of the infinite path by proving that the correct value is Rd(𝑃∞) = (12+√8)/17≈0.87226.
Theorem 1.4.4 (Corsten–DeBiasio–Lamaison–Lang [30]). There exists a2 -edge-colouring of 𝐾
N such that every monochromatic path has upper density at most
(12+√8)/17.
Theorem 1.4.5(Corsten–DeBiasio–Lamaison–Lang [30]). Suppose the edges of𝐾
N are coloured with two colours. Then, there exists a monochromatic path with upper density at least (12+√8)/17.
Note that the problem of determining the𝑟-colour Ramsey upper density remains
open and it would be very interesting to make any improvement on Rado’s lower bound of 1/𝑟 for 𝑟 ≥ 3 colours (see [33, Corollary 3.5] for the best known upper
bound). In particular for three colours, the correct value is between 1/3 and 1/2.
Question 1.4.6. What isRd𝑟(𝑃∞) for𝑟 ≥ 3?
The proof of Theorems1.4.4and1.4.5, which is based on a mix of the regularity method and novel yet elementary combinatorial arguments, will be presented in Section4.1.
Bipartite Ramsey densities
In Section4.2.5we will briefly discuss a bipartite version of Theorem1.4.5. Gyárfás and Lehel [61] and independently Faudree and Schelp [51] proved that every 2-edge-coloured𝐾𝑛,𝑛 contains a monochromatic path with at least 2b𝑛/2c +1 vertices (that
is, roughly half the vertices of the graph). They further proved that this is best possible. We will prove an analogue of this for infinite graphs. Here, 𝐾
N,N is the infinite complete bipartite graph with one part being all even positive integers and the other part being all odd positive integers.
Theorem 1.4.7 (Corsten–DeBiasio–McKenney). Every 2-coloured 𝐾
N,N contains a monochromatic path of upper density at least1/2.
Pokrovskiy [95] proved that the vertices of every 2-edge-coloured complete bi-partite graph𝐾𝑛,𝑛can be partitioned into three monochromatic paths. Soukup [112]
proved an analogue of this for infinite graphs which holds for multiple colours: the vertices of every𝑟-edge-coloured𝐾
N,Ncan be partitioned into 2𝑟−1 monochromatic paths. He also presents an example where this is best possible. However, in his example all but finitely many vertices can be covered by𝑟 monochromatic paths.
Our next result shows that this is always possible in the case of two colours.
Theorem 1.4.8 (Corsten–DeBiasio–McKenney). The vertices of every 2 -edge-coloured 𝐾
N,N can be partitioned into a finite set and at most two monochromatic paths.
Theorem1.4.7is an immediate consequence of Theorem1.4.8. We will provide an example which demonstrates that Theorems1.4.7and1.4.8are best possible (see Example4.2.9). We believe that a similar statement is true for multiple colours. Conjecture 1.4.9 (Corsten–DeBiasio–McKenney). The vertices of every 𝑟
-edge-coloured 𝐾
N,N can be partitioned into a finite set and at most 𝑟 monochromatic paths.
Example4.2.9further shows that Conjecture1.4.9is best possible, if true.
1.4.2 Infinite Trees
In Section4.2.6we will present an analogue of Conjecture1.2.3(which implies that every 2-edge-coloured𝐾𝑛contains a monochromatic copy of every tree on𝑛/2+1
vertices) for infinite trees.
Theorem 1.4.10 (Corsten–DeBiasio–McKenney). Rd(𝑇) ≥ 1/2for every infinite
tree𝑇.
There are a couple of examples showing that this is best possible. For example, we have Rd(𝑆∞) = 1/2, where 𝑆∞ is the infinite star (that is the infinite tree in
which one vertex has infinitely many neighbours of degree 1). This can be seen by colouring the edges of𝐾
Nuniformly at random with red and blue (see Rado colouring Section4.2.2). Another such example is𝑇∞, the infinite tree in which every vertex
has infinite degree (see Example4.2.8). Furthermore, there is a locally finite (that is, every vertex has finite degree) tree𝑇 with Rd(𝑇) =1/2 (see Example4.2.8).
1.4.3 Infinite Graphs with “Linear Ramsey Number”
In the finite case, the study of graphs with linear Ramsey number has received a lot of attention and we will now study possible analogues for infinite graphs. Recall that a sequence of graphs F = {𝐹1, 𝐹2, . . .} has linear 𝑟-colour Ramsey number
if and only if there is some 𝛿 > 0 such that every 𝑟-edge-coloured 𝐾𝑛 contains a
monochromatic copy of𝐹𝑚 for every𝑚 ≤ 𝛿𝑛.
