# The phenomenon of revivals on complex potential Schrödinger’s equation

L. Boultonβ, G. Farmakisφ, B. Pelloniπ Heriot-Watt University & Maxwell
Institute for the Mathematical Sciences, Edinburgh, Scotland.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: 3rd April 2024)

###### Abstract.

The mysterious phenomena of revivals in linear dispersive periodic equations
was discovered first experimentally in optics in the 19th century, then
rediscovered several times by theoretical and experimental investigations.
While the term has been used systematically and consistently by many authors,
there is no consensus on a rigorous definition. In this paper, we describe
revivals modulo a regularity condition in a large class of Schrödinger’s
equations with complex bounded potentials. As we show, at rational times the
solution is given explicitly by finite linear combinations of translations and
dilations of the initial datum, plus an additional continuous term.

##  1\. Introduction

Recently there have been significant developments in the study of revivals in
dispersive evolution equations [18]. These phenomena, which are also called
dispersive quantisations or Talbot effects, describe a surprising dichotomy in
the pointwise behaviour of the solution of time-evolution equations at
specific values of the time variable, the so-called rational times, compared
to all other generic times. At rational times, the solution revives the shape
of the initial datum by finite superpositions, reflections and re-scalings,
with a prescribed simple combinatorial rule. See [9] and references therein.

The majority of past investigations about revivals involve equations and
boundary conditions with the property that the eigenpairs of the spatial
operator can be found explicitly and satisfy precise conditions of modularity
and periodicity. The prime example of this is the case of linear dispersive
equations with constant coefficients and periodic boundary conditions. In such
cases, the techniques for detecting the times at which the revivals appear
exploit the specific periodic matching of the eigenvalues, eigenfunctions and
boundary conditions. With the help of summations of Gauss type, the infinite
series representation of the solution then reduces to a finite sum,
characterising the revivals explicitly. Naturally, the direct applicability of
this approach is limited.

The purpose of this paper is to take a different point of view, by formulating
the revivals phenomena in terms of perturbation theory. Concretely, we ask the
question of whether a (large) class of equations exhibit them, modulo a
regular perturbation. Earlier version of this concept can be found in the
works [17, 6, 8, 7], about which we give details below.

We consider the class of linear Schrödinger equations

$\displaystyle\partial_{t}u(x,t)=-i(-\partial_{x}^{2}+V(x))u(x,t),$
$\displaystyle x\in(0,\pi),\;t>0,$ (1) $\displaystyle u(0,t)=u(\pi,t)=0,$
$\displaystyle t>0,$ $\displaystyle u(x,0)=f(x),$ $\displaystyle x\in(0,\pi),$

with a complex-valued potential $V$, subject to Dirichlet boundary conditions,
given an initial wavefunction $f\in L^{2}(0,\pi)$. Our goal is to detect
revivals by perturbation from the case $V=0$. As we shall see below, in the
large wavenumber asymptotic regime and for small enough $V$, the simple but
non-trivial structure of (1) supports the combinatorial argument, involving
the Gauss summations, that is valid for the free-space equation.

Our contribution is summarised in the next theorem. It shows that the solution
of (1) at rational times support revivals modulo a continuous term. This
result matches a similar earlier finding, reported in [2]. Indeed, for $V=0$
and boundary conditions of the type $bu(0,t)=(1-b)\partial_{x}u(\pi,t)$ where
$b\in(0,1)$ is a parameter, an analogous conclusion holds true.

Here and everywhere below, $f^{\mathrm{o}}$ denotes the odd, $2\pi$-periodic
extension of the function $f$ and $\langle
V\rangle=\frac{1}{\pi}\int_{0}^{\pi}V(x)\mathrm{d}x$ the mean of the potential
function.

######  Theorem 1.

Let $V\in H^{2}(0,\pi)$ with $\|V\|_{\infty}<\frac{3}{2}$. Then, for
$p,q\in\mathbb{N}$ co-prime numbers, the solution $u(x,t)$ to (1) at time
$t=2\pi\frac{p}{q}$ is given by

$u\Big{(}x,2\pi\frac{p}{q}\Big{)}=w\Big{(}x,2\pi\frac{p}{q}\Big{)}+\frac{1}{q}\
e^{-2\pi i\langle V\rangle\frac{p}{q}}\sum_{k,m=0}^{q-1}e^{2\pi
i(m\frac{k}{q}-m^{2}\frac{p}{q})}f^{\mathrm{o}}\Big{(}x-2\pi\frac{k}{q}\Big{)}$

where $w(\cdot,t)\in\operatorname{C}(0,\pi)$ for all fixed $t>0$.

In case $V$ is real-valued, the same conclusion holds with the weaker
assumptions $V\in\operatorname{BV}(0,\pi)$ and $\|V\|_{\infty}<\infty$. In
Section 4 we investigate whether the bound on $V$ is necessary in the complex
case.

We interpret the conclusion of Theorem 1 by saying that (1) supports a weak
form of revival. Note that the result does not depend on the orthogonality of
the family of eigenfunctions. Our choice of Dirichlet boundary conditions,
corresponding to potential barriers at both ends of the finite segment,
exhibits all the features of our methodology but is free from unnecessary
complications. Indeed, while Neumann and other separated boundary conditions
can be treated via a similar approach, in these cases multiplicities and the
possibility of eigenvalues which are not semi-simple lead to additional
technical distractions.

Phenomena similar to the one described here had already been observed. The
investigations conducted in [17] and [6] lead to versions of Theorem 1 for
periodic $V$ and periodic boundary conditions. The method of proof in these
works is different from the one below. It relies on Duhamel’s representation
and an analysis of the solution in periodic Besov spaces. In a separate
development, for the periodic cubic non-linear Schrödinger equation [8] and
the Korteweg-de Vries equation [7], it was proved that at all times the
difference between the solution and the linear time-evolution is more regular
than the initial datum. This directly implies the appearence of weak revivals
at rational times also for these two non-linear equations. Numerical evidence
of this effect in the non-linear setting was reported in [4, 5]. Our findings
complement all these investigations.

The proof of Theorem 1 that we present shows how to apply directly classical
perturbation expansions in order to derive existence of revivals. This
approach has two main implications worth mentioning. On the one hand, Theorem
1 confirms the conjecture that the revival effect is prevalent in a large
class of quantum systems with discrete spectra, when the eigenpairs asymptotic
supports it [3, Section 6.2, page 116], irrespective of whether the underlying
operator is self-adjoint. On the other hand, the present approach might
provide a rigorous foundation for tackling the general conjecture formulated
in [4, page 12-13]. The latter states that a linear PDE with a dispersion
relation that is asymptotic to a polynomial with integer coefficients, in the
large wave-numbers regime, should support a type of revival. Further numerical
evidence strengthening the validity of this conjecture can be found in [16]
for the case $V=0$ and various classes of boundary conditions.

The structure of the paper is as follows. In Section 2 we lay down the precise
eigenpairs asymptotics, in terms of $V$, that allow the validity of Theorem 1.
All the results that we present in that section are classical, but we include
crucial details of their proofs. Section 3 is devoted to the proof of Theorem
1. In the final Section 4, we illustrate our main results by means of examples
involving complex Mathieu potentials and discuss the optimality of the
different assumptions on $V$.

##  2\. Spectral asymptotics

Let $V\in H^{2}(0,\pi)$ be a complex-valued potential function. Denote the
Hamiltonian associated to (1) by

$L=-\partial_{x}^{2}+V:H^{2}(0,\pi)\cap H^{1}_{0}(0,\pi)\longrightarrow
L^{2}(0,\pi).$

Since

$\|V\|_{\infty}=\max_{x\in[0,\pi]}|V(x)|<\infty,$

then the operator $L$ is closed in the domain above, it has a compact
resolvent and its adjoint $L^{*}=-\partial_{x}^{2}+\overline{V}$ has the same
domain. In some of the statements below we impose the extra condition
$\|V\|_{\infty}<\frac{3}{2}$ of the theorem. Moreover, without loss of
generality, we assume in what follows that $\langle V\rangle=0$.

The boundary value problem (1) can be written concisely as

(2) $\displaystyle u_{t}=-iLu,$ $\displaystyle u(\cdot,0)=f,$

for an initial value $f\in L^{2}(0,\pi)$. We know from the classical theory of
perturbations of one-parameter semigroups that the operator $iL$ is the
generator of a $C_{0}$ one-parameter semigroup so the equation has a unique
solution in $L^{2}$ for any $f\in L^{2}$. In general, the spectrum of $L$ is
not real. However, it is asymptotically close to the real line and the
eigenfunctions are asymptotically close to trigonometric functions. Our
objective in this section is to determine the leading order of these
asymptotics and the precise decay rate of the corrections, for the appearence
of weak revivals in (2). We then give the proof of Theorem 1 in the next
section.

The next two lemmas are routine consequences of classical properties of non-
self-adjoint Sturm-Liouville operators and their analytic perturbation theory,
but we give full details of their validity as they are not standard. They
imply that the operator $L$ has an infinite sequence of eigenpairs
(eigenfunctions and eigenvalues)

$\\{y_{j},w_{j}^{2}\\}_{j=1}^{\infty}\subset(H^{2}\cap
H^{1}_{0})\times\mathbb{C}$

with an asymptotic structure close enough to that of the case $V=0$.

######  Lemma 1.

Let $w_{j}^{2}$ be the eigenvalues of $L$. Then, for $j\to\infty$,

$|w_{j}-j|=\frac{a_{3}}{j^{3}}+O(j^{-4})$

where $a_{3}\in\mathbb{C}$ is a constant that only depends on $V$. Moreover,
if $\|V\|_{\infty}<\frac{3}{2}$, then each eigenvalue $w_{j}^{2}$ is simple.

###### Proof.

By virtue of the classical Marchenko asymptotic formula [14, Theorem 1.5.1],
we know that

$w_{j}=j+\frac{a_{1}}{2j}+\frac{\tilde{a}_{3}}{8j^{3}}+O(j^{-4})\qquad
k\to\infty$

where $a_{1}=\langle V\rangle=0$. This gives the eigenvalue asymptotics.

The family of operators $\alpha\longmapsto T_{\alpha}=-\partial_{x}^{2}+\alpha
V$ on the domain $H^{2}\cap H^{1}_{0}$ is a holomorphic family of type (A) for
$\alpha\in\mathbb{C}$; see [13, Example 2.17, p.385]. As the operator $T_{0}$
has a compact resolvent, then it follows that $T_{\alpha}$ have compact
resolvent for all $\alpha\in\mathbb{C}$.

Now, assume that $\|V\|_{\infty}<\frac{3}{2}$. Then, for $|\alpha|\leq 1$, all
the eigenvalues of the family $T_{\alpha}$ are simple, as they lie in the
$\alpha\|V\|_{\infty}$-neighbourhood of $\\{j^{2}\\}_{j=1}^{\infty}$. In
particular, for $\alpha=1$ all the eigenvalues of $L$ are indeed simple. ∎

We now show that the eigenfunctions of $L$ are close enough to the orthonormal
Fourier-sine basis corresponding to $V=0$,

(3) $d_{j}(x)=\sqrt{\frac{2}{\pi}}\sin(j\pi x),\quad j\in\mathbb{N}.$

From the second statement of the next lemma, it follows that the solution to
(2) is given by

$u(x,t)=\sum_{j=1}^{\infty}\langle f,y_{j}^{*}\rangle
e^{-iw_{j}^{2}t}y_{j}(x)$

for $y_{j}^{*}$ the eigenfunctions of $L^{*}$ scaled to form a bi-orthogonal
set paired with $y_{j}$, $\langle y_{j},y_{k}^{*}\rangle=\delta_{jk}$, and the
series converges in $L^{2}$. This turns out to be crucial for the validity of
Theorem 1.

######  Lemma 2.

In the asymptotic regime $j\to\infty$, the eigenfunctions of $L$ are such that

(4) $y_{j}(x)=c\left[\sin(jx)-\frac{\cos(jx)V_{1}(x)}{2j}+R_{j}(x)\right]$

where $\|R_{j}\|_{\infty}=O(j^{-2})$ and $V_{1}$ is defined by

(5) $V_{1}(x)=\int_{0}^{x}V(s)\,\mathrm{d}s.$

If $\|V\|_{\infty}<\frac{3}{2},$ then

$\left\\{\frac{y_{j}}{\|y_{j}\|_{2}}\right\\}_{j=1}^{\infty}$

is a Riesz basis for $L^{2}(0,\pi)$.

###### Proof.

According to [14, Lemma 1.4.1], the eigenfunction associated to $w_{k}$ is

$y_{j}(x)=A^{+}\Phi_{j}^{+}(x)+A^{-}\Phi_{j}^{-}(x)$

where

$\Phi_{j}^{\pm}(x)=e^{\pm
iw_{j}x}\left(1\pm\frac{V_{1}(x)}{2iw_{j}}+\frac{V_{2}(x)}{(2iw_{j})^{2}}+R_{j}^{\pm}(x)\right)$

for $\|R^{\pm}_{j}\|_{\infty}=O(w_{j}^{-3})$, $V_{1}$ given by (5) and

$\displaystyle\begin{aligned} V_{2}(x)&=\int_{0}^{x}LV_{1}(s)\,\mathrm{d}s\\\
&=\int_{0}^{x}V(s)V_{1}(s)\,\mathrm{d}s-V(x)+V(0).\end{aligned}$

Moreover, again from [14, Lemma 1.4.1], we know that $R^{\pm}_{j}(0)=0$. Thus,
substituting the boundary conditions $y_{j}(0)=0$, we get $-A^{-}=A^{+}$. Set
the latter equal to 1. Hence,

$\displaystyle
y_{j}(x)=\sin(w_{j}x)-\frac{\cos(w_{j}x)V_{1}(x)}{2w_{j}}+\tilde{R}_{j}(x)\quad\text{where}$
$\displaystyle\quad\|\tilde{R}_{j}\|_{\infty}=O(j^{-2}).$

Now,

$\displaystyle\sin\big{(}jx+O(j^{-3})x\big{)}$
$\displaystyle=\sin(jx)+s_{j}(x)\qquad\text{and}$
$\displaystyle\cos\big{(}jx+O(j^{-3})x\big{)}$
$\displaystyle=\cos(jx)+c_{j}(x)$

where

$\displaystyle|s_{j}(x)|+|c_{j}(x)|$ $\displaystyle\leq
2\big{|}\cos(O(j^{-3})x)-1\big{|}+2\big{|}\sin(O(j^{-3})x)\big{|}$
$\displaystyle\leq k_{1}j^{-6}+k_{2}j^{-3}$

for all $x\in[0,\pi]$. This gives (4), by taking
$R_{j}(x)=\tilde{R}_{j}(x)+s_{j}(x)+c_{j}(x)$.

Let us now show that if $\|V\|_{\infty}<\frac{3}{2}$, then the eigenfunctions
form a Riesz basis. We aim at applying [13, Theorem 2.20, p.265]. According to
[14, Theorem 1.3.1] combined with Lemma 1, the family of eigenfunctions
$\\{y_{j}\\}$ is complete in $L^{2}(0,\pi)$. Since it has a dual pair,
$\\{y^{*}_{j}\\}$, then it is minimal and so therefore exact [11]. Minimality
ensures that $\\{y_{j}\\}$ is $\omega$-independent [11]. This gives two of the
hypotheses of [13, Theorem 2.20, p.265].

Now, by virtue of (4) already proven, there exists a constant $c_{4}>0$ such
that

$\left\|\frac{y_{j}}{\|y_{j}\|_{2}}-d_{j}\right\|_{2}\leq\frac{k_{3}}{j}.$

Thus,

$\sum_{j=1}^{\infty}\left\|\frac{y_{j}}{\|y_{j}\|_{2}}-d_{j}\right\|_{2}^{2}<\infty.$

This is the other hypothesis required in [13, Theorem 2.20, p.265] and so
indeed111The conclusion of [13, Theorem 2.20, p.265] does not exactly state
that the family is a Riesz basis. But the proof implies that the family is
equivalent to an orthonormal basis, hence it is indeed always a Riesz basis.
$\left\\{\frac{y_{j}}{\|y_{j}\|}\right\\}_{j=1}^{\infty}$ is a Riesz basis of
$L^{2}(0,\pi)$. ∎

We discuss the optimality of the condition $\|V\|_{\infty}<\frac{3}{2}$ and
the case where $V$ is real-valued in Section 4.

From the above, we gather that the eigenvalues of $L$ are

(6) $\lambda_{j}=w_{j}^{2}=j^{2}+\frac{k_{j}}{j^{2}}\qquad\text{for suitable
}\\{k_{j}\\}\in\ell^{\infty}.$

Let

$n_{j}(x)=\sqrt{\frac{2}{\pi}}\cos(j\pi x),\quad j\in\mathbb{N}$

be the non-constant orthonormal Fourier-cosine basis. Below we fix the
eigenfunctions according to a normalisation of their bi-orthogonal pairs.
Concretely, let

(7)
$\phi_{j}(x)=\gamma_{j}y_{j}(x)=\gamma_{j}d_{j}(x)-\frac{\gamma_{j}n_{j}(x)V_{1}(x)}{2j}+\gamma_{j}R_{j}(x)$

where $\|R_{j}\|_{\infty}\leq\frac{c}{j^{2}}$. Without further mention, from
now on the non-zero constants $\gamma_{j}$ are chosen, such that the
associated bi-orthogonal sequence $\\{\phi^{*}_{j}\\}$ is given by

(8)
$\phi_{j}^{*}(x)=d_{j}(x)-\frac{n_{j}(x)\overline{V}_{1}(x)}{2j}+R_{j}^{*}(x).$

Then, there exist constants $0<\tilde{\gamma}_{1}<\tilde{\gamma}_{2}<\infty$
such that

(9)
$\frac{\tilde{\gamma}_{1}}{j}<|\gamma_{j}-1|<\frac{\tilde{\gamma}_{2}}{j}\qquad\text{and}\qquad\frac{\tilde{\gamma}_{1}}{j}<\big{|}\|\phi^{*}_{j}\|_{2}-1\big{|}<\frac{\tilde{\gamma}_{2}}{j},$

for all $j\in\mathbb{N}$.

##  3\. Proof of Theorem 1

We now state and prove a crucial lemma, from which Theorem 1 follows as a
corollary.

######  Lemma 3.

If the potential $V$ is such that $\langle V\rangle=0$ and
$\|V\|_{\infty}<\frac{3}{2}$, then the solution to the time-evolution equation
(2) is given by

(10) $u(x,t)=w(x,t)+\sum_{j=1}^{\infty}\langle f,d_{j}\rangle
e^{-ij^{2}t}d_{j}(x),$

where for each fixed $t>0$, $w(\cdot,t)\in C([0,\pi])$.

###### Proof.

We separate the proof into four steps.

Step 1. Consider the $L^{2}$ expansion of the initial condition,

$f(x)=\sum_{j=1}^{\infty}c_{j}\phi_{j}(x),\quad c_{j}=\langle
f,\phi^{*}_{j}\rangle.$

According to (6),

$\displaystyle u(x,t)$
$\displaystyle=\sum_{j=1}^{\infty}c_{j}e^{-i\lambda_{j}t}\phi_{j}(x)=\sum_{j=1}^{\infty}c_{j}e^{-i\left(j^{2}+\frac{k_{j}}{j^{2}}\right)t}\phi_{j}(x)$
$\displaystyle=\sum_{j=1}^{\infty}c_{j}e^{-ij^{2}t}\left(1-\frac{ik_{j}}{j^{2}}\int_{0}^{t}e^{-\frac{ik_{j}}{j^{2}}s}\,\mathrm{d}s\right)\phi_{j}(x)$
$\displaystyle=U_{1}(x,t)-U_{2}(x,t)$

where

$U_{1}(x,t)=\sum_{j=1}^{\infty}c_{j}e^{-ij^{2}t}\phi_{j}(x)$

and $U_{2}(x,t)$ has a similar expression but involving the integral above. We
treat these two terms separately.

Step 2. Let us show that $U_{2}\in C^{1}([0,\pi])$. Set

$\zeta_{j}(x,t)=\frac{ic_{j}k_{j}}{j^{2}}e^{-ij^{2}t}\int_{0}^{t}e^{-\frac{ik_{j}}{j^{2}}s}\,\mathrm{d}s\
\phi_{j}(x).$

Then,

$|\zeta_{j}(x,t)|\leq\frac{\|\\{k_{j}\\}\|_{\infty}\|\phi_{j}\|_{\infty}|\langle
v,\phi_{j}^{*}\rangle|}{j^{2}}tB_{j}\leq\frac{\|\\{k_{j}\\}\|_{\infty}\|\phi_{j}\|_{\infty}\|v\|_{2}\|\phi_{j}^{*}\|_{2}}{j^{2}}tB_{j}$

where

$B_{j}=\sup_{s\in[0,t]}\left|e^{-\frac{ik_{j}s}{j^{2}}}\right|\leq\sup_{s\in[0,t]}e^{\left|\frac{\operatorname{Im}k_{j}s}{j^{2}}\right|}\leq
B<\infty.$

According to (6), the right hand side constant is independent of $j$. Here $t$
is fixed. Moreover, by virtue of (7) and (9),

$\max\big{\\{}\|\phi^{*}_{j}\|_{2},\|\phi_{j}\|_{\infty}\big{\\}}\leq c$

for all $j\in\mathbb{N}$, where the constant $c>0$ is independent of $j$.
Hence, by Weierstrass’ M-test,

$U_{2}(x,t)=\sum_{j=1}^{\infty}\zeta_{j}(x,t)$

converges absolutely and uniformly to a $C^{1}$ function, because each
component is $C^{1}$.

Step 3. Consider now $U_{1}$. According to (7),

$\displaystyle U_{1}(x,t)$
$\displaystyle=\sum_{j=1}^{\infty}c_{j}\gamma_{j}e^{-ij^{2}t}d_{j}(x)-\sum_{j=1}^{\infty}\frac{c_{j}\gamma_{j}e^{-ij^{2}t}}{2j}n_{j}(x)V_{1}(x)+\sum_{j=1}^{\infty}c_{j}\gamma_{j}e^{-ij^{2}t}R_{j}(x)$
$\displaystyle=u_{3}(x,t)+u_{4}(x,t)+u_{5}(x,t).$

In this step we show that $u_{4}$ and $u_{5}$ are continuous in the variable
$x$. From the asymptotic behaviour of $R_{j}$ and an identical argument as we
used in step 2, we know that $u_{5}$ is $C^{1}$ in the variable $x$.

Now, for $u_{4}(x,t)$, note that

$\sum_{j=1}^{\infty}|c_{j}|^{2}<\infty,$

because $\\{\phi_{j}^{*}\\}$ is a Riesz basis, e.g. [11, Theorem 7.13]. Then,
by Cauchy-Schwarz,

$\sum_{j=1}^{\infty}\left|\frac{c_{j}}{j}\right|<\infty.$

Thus, since $V_{1}$ is a bounded function, for all fixed $t$, the sequence

$\left\\{\frac{c_{j}\gamma_{j}e^{-ij^{2}t}\|V_{1}\|_{\infty}}{2j}\right\\}_{j=1}^{\infty}\in\ell^{1}(\mathbb{N}).$

Hence, for all fixed $t>0$, the family of sequences (family for $x\in[0,\pi]$)

$\left\\{\frac{c_{j}\gamma_{j}e^{-ij^{2}t}n_{j}(x)V_{1}(x)}{2j}\right\\}_{j=1}^{\infty}\in\ell^{1}(\mathbb{N}).$

Therefore, by the Dominated Convergence Theorem (in $\ell^{1}$), we have that

$\lim_{x\to x_{0}}u_{4}(x,t)=u_{4}(x_{0},t),$

for all $x_{0}\in[0,\pi]$. That is, $u_{4}$ is a continuous function of the
variable $x$.

Step 4. Finally, we consider $u_{3}(x,t)$. By (8), we know that

$c_{j}=\langle f,d_{j}\rangle-\frac{\langle fV_{1},n_{j}\rangle}{2j}+\langle
f,R_{j}^{*}\rangle.$

Then, $u_{3}(x,t)=u_{6}(x,t)-u_{7}(x,t)+u_{8}(x,t),$ where

$\displaystyle u_{6}(x,t)=\sum_{j=1}^{\infty}\langle
f,d_{j}\rangle\gamma_{j}e^{-ij^{2}t}d_{j}(x),\qquad
u_{7}(x,t)=\sum_{j=1}^{\infty}\frac{\langle
fV_{1},n_{j}\rangle}{2j}\gamma_{j}e^{-ij^{2}t}d_{j}(x)$
$\displaystyle\text{and}\quad u_{8}(x,t)=\sum_{j=1}^{\infty}\langle
f,R_{j}^{*}\rangle\gamma_{j}e^{-ij^{2}t}d_{j}(x).$

We write $\gamma_{j}=1+(\gamma_{j}-1)$ and split each term of $u_{6}(x,t)$,
$u_{7}(x,t)$, $u_{8}(x,t)$ into two sums.

For $u_{6}(x,t)$ we have

$u_{6}(x,t)=\sum_{j=1}^{\infty}\langle f,d_{j}\rangle
e^{-ij^{2}t}d_{j}(x)+\sum_{j=1}^{\infty}\langle
f,d_{j}\rangle(\gamma_{j}-1)e^{-ij^{2}t}d_{j}(x)$

The first component of $u_{6}$ is the second component in the solution
representation (10). To deal with the second one we use (9). By Cauchy-
Schwarz,

$\sum_{j=1}^{\infty}\frac{|\langle f,d_{j}\rangle|}{j}<\infty.$

Hence, by Weierstrass’ M-test again, for each $t>0$, the series of functions

$\sum_{j=1}^{\infty}\langle f,d_{j}\rangle(\gamma_{j}-1)e^{-ij^{2}t}d_{j}(x)$

converges absolutely and uniformly to a $C^{1}$ function on $[0,\pi]$.

Now, the function $u_{7}$ is written as follows

$u_{7}(x,t)=\sum_{j=1}^{\infty}\frac{\langle
fV_{1},n_{j}\rangle}{2j}e^{-ij^{2}t}d_{j}(x)+\sum_{j=1}^{\infty}\frac{\langle
fV_{1},n_{j}\rangle}{2j}(\gamma_{j}-1)e^{-ij^{2}t}d_{j}(x).$

The first component of $u_{7}$ is continuous as a consequence of an argument
similar to that employed for $u_{4}$. Indeed, since $\\{n_{j}\\}$ is an
orthonormal basis and $fV_{1}\in L^{2}(0,\pi)$,

$\sum_{j=1}^{\infty}\left|\frac{\langle fV_{1},n_{j}\rangle
e^{-ij^{2}t}}{2j}\right|\leq\|fV_{1}\|_{2}\frac{\pi}{\sqrt{6}},$

so we can use the sequence

$\left\\{\frac{\langle fV_{1},n_{j}\rangle
e^{ij^{2}t}}{2j}\right\\}_{j=1}^{\infty}\in\ell^{1}$

to ensure continuity, via the Dominated Convergence Theorem. The second
component is a $C^{1}$ function since its Fourier-sine coefficients decay like
$j^{-2}$, due to (9) and the Cauchy-Schwarz inequality. So, in total,
$u_{7}(\cdot,t)\in C([0,\pi])$.

Finally, for the function $u_{8}$ we have that

$u_{8}(x,t)=\sum_{j=1}^{\infty}\langle f,R_{j}^{*}\rangle
e^{-ij^{2}t}d_{j}(x)+\sum_{j=1}^{\infty}\langle
f,R_{j}^{*}\rangle(\gamma_{j}-1)e^{-ij^{2}t}d_{j}(x).$

Here, the first component is $C^{1}$ in $x$. Indeed, since

$|\langle
f,R_{j}^{*}\rangle|=\left|\int_{0}^{\pi}f(x)\overline{R_{j}^{*}(x)}\,\mathrm{d}x\right|\leq\frac{\pi\|f\|_{2}c}{j^{2}},$

the first component is a function whose sine-Fourier coefficients decay like
$j^{-2}$, so it is continuously differentiable in $x$. The second component is
twice differentiable in $x$, since due to (9) it represents a function whose
Fourier-sine coefficients decay like $j^{-3}$. Thus, $u_{8}(\cdot,t)$ belongs
to $C^{1}([0,\pi])$.

Collecting the statements about $u_{k}$ from the previous steps, we conclude
that the expression of $u$ is as claimed in (10) where indeed $w$ is
continuous in the variable $x$. ∎

######  Remark 1.

In the proof of this lemma, note that all components of $w$ are $C^{1}$,
except $u_{4}$ and $u_{7}$.

The proof of Theorem 1 now follows from Lemma 3 and the combinatorial argument
for $V=0$.

###### Proof of Theorem 1.

We show that, if $V=0$, then the solution to (2) at rational times
$t_{\mathrm{r}}=2\pi\frac{p}{q}$ is

(11) $u(x,t_{\mathrm{r}})=\sum_{j=1}^{\infty}\langle f,d_{j}\rangle
e^{-ij^{2}t_{\mathrm{r}}}d_{j}(x)=\frac{1}{q}\sum_{k,m=0}^{q-1}e^{2\pi
i(-m^{2}\frac{p}{q}+m\frac{k}{q})}f^{\mathrm{o}}\Big{(}x-2\pi\frac{k}{q}\Big{)},$

where $f^{\mathrm{o}}$ denotes the odd, $2\pi$-periodic extension of $f$.
Therefore, replacing $V$ with $V-\langle V\rangle$ if needed, and applying
Lemma 3, gives Theorem 1.

The proof of (11) is as follows. For $t\in\mathbb{R}$ we have that

$u(x,t)=\frac{1}{2\pi}\sum_{j=-\infty}^{\infty}e^{-ij^{2}t}\langle
f^{\mathrm{o}},e^{ij(\cdot)}\rangle e^{ijx}.$

Then, for $t=t_{\mathrm{r}}$ take $j\underset{q}{\equiv}m$ so that
$e^{ij^{2}t_{\mathrm{r}}}=e^{im^{2}t_{\mathrm{r}}}$. Thus,

$u(x,t_{\mathrm{r}})=\frac{1}{2\pi}\sum_{m=0}^{q-1}e^{-im^{2}t_{\mathrm{r}}}\sum_{\begin{subarray}{c}j\in\mathbb{Z}\\\
j\underset{q}{\equiv}m\end{subarray}}\langle
f^{\mathrm{o}},e^{ij(\cdot)}\rangle e^{ijx}.$

Let the summation on the right had side be denoted by $T$. Since

$\sum_{k=0}^{q-1}e^{2\pi
i(m-j)\frac{k}{q}}=\begin{cases}q&j\underset{q}{\equiv}m\\\
0&j\underset{q}{\not\equiv}m,\end{cases}$

we have

$\displaystyle T$ $\displaystyle=\frac{1}{q}\sum_{k=0}^{q-1}e^{2\pi
im\frac{k}{q}}\sum_{j\in\mathbb{Z}}e^{-2\pi i\frac{k}{q}j}\langle
f^{\mathrm{o}},e^{ij(\cdot)}\rangle e^{ijx}$
$\displaystyle=\frac{1}{q}\sum_{k=0}^{q-1}e^{2\pi
im\frac{k}{q}}\sum_{j\in\mathbb{Z}}\Big{\langle}f^{\mathrm{o}}\Big{(}\cdot-\frac{2\pi
k}{q}\Big{)},e^{ij(\cdot)}\Big{\rangle}e^{ijx}$
$\displaystyle=\frac{1}{q}\sum_{k=0}^{q-1}e^{2\pi
im\frac{k}{q}}f^{\mathrm{o}}\Big{(}x-\frac{2\pi k}{q}\Big{)}.$

Hence, (11) holds true. ∎

######  Remark 2.

For general bounded complex potential $V$, we know from the Dyson expansion
that (10) holds true for

$w(x,t)=\sum_{k=1}^{\infty}w_{k}(x,t),$

where $w_{k}(x,t)$ are explicitly given in terms of integrals of regular
functions. These functions are continuous for $x\in(0,\pi)$. By tracking the
convergence of the series, it might be possible to establish its continuity,
therefore extend the results of this section to all $\|V\|_{\infty}<\infty$.
See the ideas described in [1, §7].

##  4\. The hypotheses of Theorem 1

In this section we examine the optimality of the hypotheses of Theorem 1, in
terms of the size and the regularity of the potential.

Firstly, we note that for $L$ self-adjoint, the assumptions on $V$ can be
relaxed. According to the asymptotic expansions reported in [10, eqs.
$(4.21)_{7}$, $(5.4)_{2}$ and $(5.9)_{2}$], the identity (4) is valid for any
$V:[0,\pi]\longrightarrow\mathbb{R}$ of bounded variation irrespective of the
size of $\|V\|_{\infty}<\infty$. Therefore, by following the same method of
proof presented above, it directly follows that the conclusion of Theorem 1
still holds true under these modified hypotheses on $V$. That is, the solution
to (1) at rational times, is given by

(12)
$u\Big{(}x,2\pi\frac{p}{q}\Big{)}=w\Big{(}x,2\pi\frac{p}{q}\Big{)}+\frac{1}{q}\
e^{-2\pi i\langle V\rangle\frac{p}{q}}\sum_{k,m=0}^{q-1}e^{2\pi
i(m\frac{k}{q}-m^{2}\frac{p}{q})}f^{\mathrm{o}}\Big{(}x-2\pi\frac{k}{q}\Big{)}$

for a suitable function $w(\cdot,t)$ continuous in $x\in[0,\pi]$.

In the more general non-self-adjoint setting, for
$\|V\|_{\infty}>\frac{3}{2}$, we only know from the available asymptotic
formulas that, for large wavenumbers, all the eigenvalues are simple and the
corresponding eigenfunctions form a basis of a subspace $\mathcal{S}$ of
finite co-dimension. Despite of this, still $iL$ is the generator of a one-
parameter semigroup, see Remark 2 above. The solution to (1) exists and it is
unique for all $f\in L^{2}$. Moreover, (2) has a solution with an
$L^{2}$-convergent eigenfunction expansion and a version of Theorem 1 can be
recovered for all $f\in\mathcal{S}$.

We now present an example of a purely imaginary $V\in C^{\infty}$ with
$\|V\|_{\infty}=2$, for which (12) appears to still be valid. For this purpose
we choose for $V$ a purely imaginary Mathieu potential.

Let $q\in\mathbb{C}$ and $V(x)=2q\cos(2x)$. Then $\langle V\rangle=0$. The
eigenvalue equation associated to the operator $L$ is Mathieu’s equation. The
eigenvalues of $L$ are $\omega_{j}^{2}=b_{j}(q),$ the Mathieu characteristic
values, which satisfy

$b_{j}(q)=j^{2}+\frac{1}{2(j^{2}-1)}q^{2}+O(q^{4})$

as $|q|\to 0$. The corresponding eigenfunctions are the Mathieu functions

$\phi_{j}(x)=\operatorname{se}_{j}(x,q)$

for $j\in\mathbb{N}$. See [12, §7.4] and also [15, 28.2-7].

In Figure 1 we set $f(x)=\chi_{[\frac{3\pi}{8},\frac{5\pi}{8}]}(x)$ and show a
numerical approximation to 100 modes of $u(x,t)$ at time $t=\frac{2\pi}{5}$
for purely imaginary $q$ with increasing modulus. As $|q|$ increases, we show
how the correction $w(x,t)$ affects the revivals part of the solution. Note
that the conclusions of Theorem 1 only hold for $|q|<\frac{3}{4}$, but the
numerical approximation in the figure suggests that this conclusion appears to
be valid also when $\|V\|_{\infty}=2$ for this potential.

The graphs shown in Figure 2 re-inforce the conjecture that the correction
term $w$ is continuous beyond the threshold $\|V\|_{\infty}=\frac{3}{2}$.
Indeed, in Figure 2 we show a 100 modes approximation of

$u(x,t)-\sum_{j=1}^{\infty}\langle f,d_{j}\rangle e^{-ij^{2}t}d_{j}(x)$

for the same data as in Figure 1. For (a)-(b) we confirm the shape of
$w(x,t)$. For (c)-(d), note that even when $q=\frac{3\pi}{4}i$ and $q=i$, the
difference appears to still be continuous.

## Acknowledgements

We kindly thank David Smith for his valuable comments made during the
preparation of this manuscript. The work of GF was funded by EPSRC through
Heriot-Watt University support for Research Associate positions, under the
Additional Funding Programme for the Mathematical Sciences. BP was partially
supported by a Leverhulme Research Fellowship.

## References

  * [1] Lyonell Boulton and Spyridon Dimoudis, _Norm-ideal perturbations of one-parameter semigroups and applications_ , Journal of Functional Analysis 284 (2023), 109784.
  * [2] Lyonell Boulton, George Farmakis and Beatrice Pelloni, _Beyond periodic revivals for linear dispersive PDEs_ , Proceedings of the Royal Society A 477 (2021), no. 2251, 20210241.
  * [3] Siegmund Brandt and Hans D. Dahmen, _The Picture Book of Quantum Mechanics_ , Springer New York, 2012.
  * [4] Gong Chen and Peter J. Olver, _Dispersion of discontinuous periodic waves_ , Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469 (2013), no. 2149, 20120407.
  * [5] Gong Chen and Peter J Olver, _Numerical simulation of nonlinear dispersive quantization_ , Discrete & Continuous Dynamical Systems-A 34 (2014), no. 3, 991.
  * [6] Gunwoo Cho, Jimyeong Kim, Seongyeon Kim, Yehyun Kwon and Ihyeok Seo, _Quantum revivals and fractality for the Schrödinger equation_ , arXiv preprint, arXiv:2105.12510 (2023).
  * [7] Mehmet B. Erdoğan and Nikolaos Tzirakis, _Global smoothing for the periodic KdV evolution_ , International Mathematics Research Notices 2013 (2013), no. 20, 4589–4614.
  * [8] by same author, _Talbot effect for the cubic non-linear Schrödinger equation on the torus_ , Mathematical Research Letters 20 (2013), no. 6, 1081–1090.
  * [9] by same author, _Dispersive partial differential equations: wellposedness and applications_ , vol. 86, Cambridge University Press, 2016.
  * [10] Charles T. Fulton and Steven A. Pruess, _Eigenvalue and eigenfunction asymptotics for regular Sturm-Liouville problems_ , Journal of Mathematical Analysis and Applications 188 (1994), no. 1, 297–340.
  * [11] Christopher Heil, _A Basis Theory Primer_ , Springer Science & Business Media, 2011.
  * [12] Edward L. Ince, _Ordinary Differential Equations_ , Dover, 1956.
  * [13] Tosio Kato, _Perturbation Theory for Linear Operators_ , vol. 132, Springer Science & Business Media, 1980.
  * [14] Vladimir Marchenko, _Sturm-Liouville Operators and Applications_ , American Mathematical Society, 1986.
  * [15] Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert and Charles W. Clark, _The NIST Handbook of Mathematical Functions_ , Cambridge University Press, 2010.
  * [16] Peter J. Olver, Natalie E. Sheils and David A. Smith, _Revivals and fractalisation in the linear free space Schrödinger equation_ , Quarterly of Applied Mathematics 78 (2020), no. 2, 161–192.
  * [17] Igor Rodnianski, _Continued fractions and Schrödinger evolution_ , Contemporary Mathematics 236 (1999), 311–323.
  * [18] David A. Smith, _Revivals and fractalization_ , Dynamical System Web (2020), 1–8.

(a)  (b)

(c)  (d)

Figure 1. Here $V(x)=2q\cos(2x)$ and $t=\frac{2\pi}{5}$. We show an
approximation of $u(x,t)$ to 100 modes. The blue curves are the solutions as
complex-valued functions of $x$, the orange curves correspond to projections
of the real and imaginary parts of these solutions, and the black curves are
the projections corresponding to the curves traced by the solutions in the
complex plane for $x\in[0,\pi]$. The figures shown match (a) $q=\frac{i}{4}$,
(b) $q=\frac{i}{2}$, (c) $q=\frac{3i}{4}$ and (d) $q=i$.

(a)  (b)

(c)  (d)

Figure 2. Here $V(x)=2q\cos(2x)$ and $t=\frac{2\pi}{5}$. We show an
approximation of $w(x,t)$ to 100 modes. The blue curves are the complex-valued
functions of $x$, the orange curves correspond to projections of the real and
imaginary parts of these, and the black curves correspond to the graphs traced
by $w(x,\frac{2\pi}{5})$ on the complex plane for $x\in[0,\pi]$. The figures
shown match (a) $q=\frac{i}{4}$, (b) $q=\frac{i}{2}$, (c) $q=\frac{3i}{4}$ and
(d) $q=i$.


with the sum converging weakly in $B(\widetilde{H}_{2})$. Since the range of
$Z^{\ast}_{\psi}$ is ${\mathbb{C}}\cdot\xi_{1}$ for any $\psi\in H_{1}$, we
have that $Z^{\ast}_{\psi}=P_{\xi_{1}}Z^{\ast}_{\psi}$ and
$Z_{\psi}=Z_{\psi}P_{\xi_{1}}$. Then

$\displaystyle{\mathbb{E}}(a\otimes
t)=\sum_{i,j\in{\mathcal{I}}}\langle\delta_{i},A\delta_{j}\rangle
V^{\ast}\hat{\pi}_{1}(Z_{i})\hat{\pi}_{1}(P_{\xi_{1}})\hat{\pi}_{2}(T)\hat{\pi}_{1}(P_{\xi_{1}})\hat{\pi}_{1}(Z^{\ast}_{j})V.$
(5.45)

Using the unitary map
$U:\widetilde{H}_{2}\rightarrow\hat{\pi}_{1}(P_{\xi_{1}})\widehat{H}$,

$\hat{\pi}_{1}(P_{\xi_{1}})\hat{\pi}_{2}(T)\hat{\pi}_{1}(P_{\xi_{1}})=\hat{\pi}_{1}(P_{\xi_{1}})UTU^{\ast}\hat{\pi}_{1}(P_{\xi_{1}}).$
(5.46)

As such,

$\displaystyle{\mathbb{E}}(a\otimes
t)=\sum_{i,j\in{\mathcal{I}}}\langle\delta_{i},A\delta_{j}\rangle\Big{(}V^{\ast}\hat{\pi}_{1}(Z_{i})U\Big{)}T\Big{(}U^{\ast}\hat{\pi}_{1}(Z^{\ast}_{j})V\Big{)}.$
(5.47)

which is identical to (5.42) for $n=1$.

Assume now that (5.42) is true for some arbitrary $n$. Then

$\displaystyle{\mathbb{E}}_{(n+1)}\Big{(}\big{(}a\otimes a_{(n)}\big{)}\otimes
t\Big{)}={\mathbb{E}}\Big{(}a\otimes{\mathbb{E}}_{(n)}\big{(}a_{(n)}\otimes
t\big{)}\Big{)}$ (5.48)
$\displaystyle\quad=\sum_{i,j\in{\mathcal{I}}}\big{\langle}\delta_{i},A\delta_{j}\big{\rangle}\Sigma_{i}\,{\mathbb{E}}_{(n)}\big{(}a_{(n)}\otimes
t\big{)}\Sigma_{j}^{\ast}$
$\displaystyle\quad=\sum_{i,j\in{\mathcal{I}}}\sum_{\bm{i},\bm{j}\in{\mathcal{I}}^{n}}\big{\langle}\delta_{i},A\delta_{j}\big{\rangle}\big{\langle}\delta_{\bm{i}},A_{(n)}\delta_{\bm{j}}\big{\rangle}\,\Sigma_{i}\Sigma_{i_{1}}\cdots\Sigma_{i_{n}}T\,\Sigma_{j_{n}}^{\ast}\cdots\Sigma_{j_{1}}^{\ast}\Sigma_{j}^{\ast}$
$\displaystyle\quad=\sum_{\bm{i},\bm{j}\in{\mathcal{I}}^{n+1}}\,\big{\langle}\delta_{\bm{i}},\big{(}A\otimes
A_{(n)}\big{)}\delta_{\bm{j}}\big{\rangle}\,\Sigma_{i_{1}}\cdots\Sigma_{i_{n+1}}T\,\Sigma_{j_{n+1}}^{\ast}\cdots\Sigma_{j_{1}}^{\ast},$

hence (5.42) holds for $n+1$ too and, as such, for any $n$. ∎

######  Remark 5.13.

Our present understanding is that not all the reconstructed states accept a
presentation in terms of operator products. Nevertheless, the following
example shows that the class of reduced data leading to factorizable states is
quite large. $\Diamond$

######  Example 5.14.

In this example we take:

$\widehat{H}=\widetilde{H}_{1}\otimes\widetilde{H}_{2},\quad\hat{\pi}_{1}(\tilde{A})=\tilde{A}\otimes\tilde{I}_{2},\quad\hat{\pi}_{2}(\tilde{T})=\tilde{I}_{1}\otimes\tilde{T}.$
(5.49)

Then $\hat{\pi}_{1}$ and $\hat{\pi}_{2}$ commute and

$\hat{\pi}_{1}(P_{\xi_{1}})\widehat{H}={\mathbb{C}}\cdot\xi_{1}\otimes\widetilde{H}_{2}\simeq\widetilde{H}_{2},$
(5.50)

and

$\displaystyle\hat{\pi}_{1}(P_{\xi_{1}})\hat{\pi}_{2}(\tilde{T})\hat{\pi}_{1}(P_{\xi_{1}})$
$\displaystyle=\hat{\pi}_{1}(P_{\xi_{1}})\big{(}P_{\xi_{1}}\otimes\tilde{T}\big{)}\hat{\pi}_{1}(P_{\xi_{1}})$
(5.51)
$\displaystyle=\hat{\pi}_{1}(P_{\xi_{1}})\big{(}U\tilde{T}U^{\ast}\big{)}\hat{\pi}_{1}(P_{\xi_{1}})$

with $U(\tilde{\chi})=\xi_{1}\otimes\tilde{\chi}$ and
$U^{\ast}(\xi_{1}\otimes\tilde{\chi})=\tilde{\chi}$ for any
$\tilde{\chi}\in\widetilde{H}_{2}$. As such, all the assumptions in Theorem
5.12 come true. Therefore, any isometry
$V:\widetilde{H}_{2}\rightarrow\widehat{H}$ supplies a generating map, via Eq.
(5.34), which accepts a presentation in therms of products of operators.
$\Diamond$

######  Remark 5.15.

The matrix product states constructed in [6, Eq. 5.4] are covered by Example
5.14. Even though no extension of the Hilbert spaces is used in [6], the
mechanism that generates the product states is the same. Indeed, the setting
there is $\widehat{H}=H_{1}\otimes H_{2}$ with ${\rm dim}\,H_{i}<\infty$,
$\hat{\pi}_{1}(A)=A\otimes I_{2}$ and $\hat{\pi}_{2}(S)=I_{1}\otimes S$. Now,
for each $\psi\in H_{1}$, let us consider:

$W_{\psi}:H_{2}\rightarrow H_{1}\otimes H_{2},\quad
W_{\psi}(\chi)=\psi\otimes\chi,$ (5.52)

with adjoint

$W^{\ast}_{\psi}:H_{1}\otimes H_{2}\rightarrow H_{2},\quad
W^{\ast}_{\psi}(\psi^{\prime}\otimes\chi)=\langle\psi,\psi^{\prime}\rangle\,\chi.$
(5.53)

Then the following identity holds:

$\hat{\pi}_{1}\big{(}|\psi\rangle\langle\psi^{\prime}|\big{)}\hat{\pi}_{2}(T)=|\psi\rangle\langle\psi^{\prime}|\otimes
T=W_{\psi}TW_{\psi^{\prime}}^{\ast},\quad\psi,\psi^{\prime}\in H_{1}.$ (5.54)

This identity is essentially equivalent to the one in (5.51) and it played the
same role in the derivation of Eq. 5.4 in [6] as the relation (5.46) played in
the derivation of the expression (5.42). $\Diamond$

As a direct consequence of Theorem 5.12, we obtain the generalization of Eq.
5.7 from [6]:

######  Corollary 5.16.

Assume the conditions from Theorem 5.12. Then the reconstructed state accepts
the following presentation:

$\displaystyle\omega\big{(}\mathfrak{i}_{n}(a_{(n)})\big{)}$
$\displaystyle=\sum_{\alpha,\beta\in{\mathbb{N}}}\bar{\omega}\big{(}B_{\alpha\gamma}\big{)}\big{\langle}\Gamma_{n}(B_{\alpha\beta}),A\,\Gamma_{n}(B_{\gamma\beta})\big{\rangle},$
(5.55)

where $\chi_{\alpha}$ and
$B_{\alpha\beta}=|\chi_{\alpha}\rangle\langle\chi_{\beta}|$ are a basis and
the corresponding system of unit matrices of $B(\widetilde{H}_{2})$ and

$\Gamma_{n}:B(H_{2})\rightarrow H_{1}^{\otimes
n},\quad\Gamma_{n}(B)=\sum_{\bm{j}\in{\mathcal{I}}^{n}}\delta_{\bm{j}}\,{\rm
Tr}\big{(}B\,\Sigma_{j_{n}}^{\ast}\cdots\Sigma_{j_{1}}^{\ast}\big{)}.$ (5.56)

###### Proof.

By combining the defining relation
$\omega\big{(}\mathfrak{i}_{n}(a_{(n)})\big{)}=\bar{\omega}\big{(}{\mathbb{E}}_{(n)}\big{(}a_{(n)}\otimes
e\big{)}\big{)}$ and (5.42), we have

$\displaystyle\omega\big{(}a_{(n)}\big{)}=\sum_{\bm{i},\bm{j}\in{\mathcal{I}}^{k}}\langle\delta_{\bm{i}},A_{(n)}\delta_{\bm{j}}\rangle\,\bar{\omega}\big{(}\Sigma_{i_{1}}\cdots\Sigma_{i_{n}}I_{2}\,\Sigma_{j_{n}}^{\ast}\cdots\Sigma_{j_{1}}^{\ast}\big{)}.$
(5.57)

We expend the operators appearing above as weakly converging sums, such as

$\Sigma_{i_{1}}\cdots\Sigma_{i_{n}}=\sum_{\alpha,\beta\in{\mathbb{N}}}\langle\chi_{\alpha},\Sigma_{i_{1}}\cdots\Sigma_{i_{n}}\chi_{\beta}\rangle\,B_{\alpha\beta}$
(5.58)

and continue as

$\displaystyle\omega\big{(}a_{(n)}\big{)}$
$\displaystyle=\sum_{\bm{i},\bm{j}\in{\mathcal{I}}^{k}}\langle\delta_{\bm{i}},A_{(n)}\delta_{\bm{j}}\rangle\sum_{\alpha,\beta,\gamma\in{\mathbb{N}}}\,\bar{\omega}\big{(}B_{\alpha\beta}B_{\beta\gamma}\big{)}$
(5.59) $\displaystyle\qquad\qquad\qquad\times\
\langle\chi_{\alpha},\Sigma_{i_{1}}\cdots\Sigma_{i_{n}}\chi_{\beta}\rangle\langle\chi_{\beta},\,\Sigma_{j_{n}}^{\ast}\cdots\Sigma_{j_{1}}^{\ast}\chi_{\gamma}\rangle$
$\displaystyle=\sum_{\bm{i},\bm{j}\in{\mathcal{I}}^{k}}\langle\delta_{\bm{i}},A_{(n)}\delta_{\bm{j}}\rangle\sum_{\alpha,\beta,\gamma\in{\mathbb{N}}}\,\bar{\omega}\big{(}B_{\alpha\gamma}\big{)}$
$\displaystyle\qquad\qquad\times\ {\rm
Tr}\Big{(}B_{\beta\alpha}\Sigma_{i_{1}}\cdots\Sigma_{i_{n}}\Big{)}{\rm
Tr}\Big{(}B_{\gamma\beta}\,\Sigma_{j_{n}}^{\ast}\cdots\Sigma_{j_{1}}^{\ast}\Big{)}$

and the statement follows.∎

## References

  * [1] I. Affleck, T. Kennedy, E. H. Lieb, H. Tasaki, Valence bond ground states in isotropic quantum antiferromagnets, Commun. Math. Phys. 115, 477-528 (1988).
  * [2] W. B. Arveson, Subalgebras of $C^{\ast}$-algebras, Acta Mathematica 123, 141–224 (1969).
  * [3] D. P. Blecher, C. Merdy, Operator algebras and their modules: An operator space approach, (Oxford University Press, Oxford, 2004).
  * [4] M. D. Choi, E. G. Effros, Injectivity and operator spaces, J. Functional Anal. 24, 156–209 (1977).
  * [5] C. Coine, Complete boundedness of multiple operator integrals, arXiv:1908.07879 (2019).
  * [6] M. Fannes, B. Nachtergaele, R. F. Werner, Finitely correlated states on quantum spin chains, Commun. Math. Phys. 144, 443-490 (1992).
  * [7] A. Kavruk, V. I. Paulsen, I. G. Todorov, M. Tomforde, Tensor products of operator systems, Journal of Functional Analysis 261, 267–299 (2011).
  * [8] A. S. Kavruk, V. I. Paulsena, I. G. Todorov, M. Tomforde, Quotients, exactness, and nuclearity in the operator system category, Advances in Mathematics 235, 321–360 (2013).
  * [9] V. Paulsen, Completely bounded maps and operator algebras, (Cambridge University Press, Cambridge, 2002).
  * [10] V. I. Paulsen, M. Tomforde, Vector spaces with an order unit, Indiana Univ. Math. J. 58, 1319–1359 (2009).
  * [11] V. I. Paulsen, I. G. Todorov, M. Tomforde, Operator system structures on ordered spaces, Proc. London Math. Soc. 102, 25–49 (2011).
  * [12] G. Pisier, Introduction to operator space theory, (Cambridge University Press, Cambridge, 2003).
  * [13] M. A. Rieffel, Concrete realizations of quotients of operator spaces, Math. Scand. 114, 205–215 (2014).
  * [14] Z. J. Ruan, Subspaces of C*-algebras, J. Funct. Anal. 76, 217-230 (1988).
  * [15] W. F. Stinespring, Positive functions on $C^{\ast}$-algebras, Proc. Amer. Math. Soc. 6, 211-216 (1955).
  * [16] D. V. Voiculescu, K. J. Dykema, A. Nica, Free random variables, (AMS, Providence, 1992).


# Ramsey upper density of infinite graph factors

József Balogh Department of Mathematics, University of Illinois at Urbana-
Champaign, Urbana, Illinois 61801, USA; and Moscow Institute of Physics and
Technology, Russian Federation. Research is partially supported by NSF Grant
DMS-1764123, Arnold O. Beckman Research Award (UIUC Campus Research Board RB
18132), the Langan Scholar Fund (UIUC), and the Simons Fellowship. Ander
Lamaison Faculty of Informatics, Marasyk University, Brno, Czech Republic.
<EMAIL_ADDRESS>Previous affiliation: Freie Universität Berlin, Berlin,
Germany. Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) under Germany’s Excellence Strategy - The Berlin Mathematics
Research Center MATH+ (EXC-2046/1, project ID: 390685689).

###### Abstract

The study of upper density problems on Ramsey theory was initiated by Erdős
and Galvin in 1993. In this paper we are concerned with the following problem:
given a fixed finite graph $F$, what is the largest value of $\lambda$ such
that every 2-edge-coloring of the complete graph on $\mathbb{N}$ contains a
monochromatic infinite $F$-factor whose vertex set has upper density at least
$\lambda$?

Here we prove a new lower bound for this problem. For some choices of $F$,
including cliques and odd cycles, this new bound is sharp, as it matches an
older upper bound. For the particular case where $F$ is a triangle, we also
give an explicit lower bound of $1-\frac{1}{\sqrt{7}}=0.62203\dots$, improving
the previous best bound of 3/5.

##  1 Introduction

Let $K_{\mathbb{N}}$ denote the complete graph on the natural numbers. Given a
subset $S\subseteq\mathbb{N}$, its upper density is defined as
$\bar{d}(S)=\limsup\limits_{t\rightarrow\infty}\frac{|S\cap[t]|}{t}$. Given a
subgraph $G\subseteq K_{\mathbb{N}}$, its upper density is
$\bar{d}(G)=\bar{d}(V(G))$.

######  Definition 1.

Let $H$ be a countably infinite graph. The Ramsey (upper) density of $H$,
denoted by $\rho(H)$, is the supremum of the values of $\lambda$ such that, in
every red-blue-coloring of the edges of $K_{\mathbb{N}}$, there exists a
monochromatic subgraph $H^{\prime}\subseteq K_{\mathbb{N}}$ isomorphic to $H$
with $\bar{d}(H^{\prime})\geq\lambda$.

This concept was first introduced by Erdős and Galvin [6], for the particular
case of the infinite path $P_{\infty}$. They proved that
$2/3\leq\rho(P_{\infty})\leq 8/9$. After some improvements on the lower bound
by DeBiasio and McKenney [4] and by Lo, Sanhueza-Matamala and Wang [9], the
Ramsey density of the infinite path was determined by Corsten, DeBiasio, Lang
and the second author in [2], as
$\rho(P_{\infty})=(12+\sqrt{8})/17=0.87226\dots$ The parameter for general $H$
was first defined by DeBiasio and McKenney [4]. Many results on the Ramsey
density of different classes of graphs $H$ can be found in [8] and [3].

We focus on the case of infinite factors. Given a finite graph $F$, the
infinite $F$-factor, denoted by $\omega\cdot F$, is the graph formed by
infinitely many pairwise disjoint copies of $F$. Several results relating to
the Ramsey density of infinite factors appear in [8].

Some of these are given in terms of a certain fixed function $f$ which is
defined as follows. The reader should be forgiven for skipping such an
unwieldy definition, as it will not be actually used in this paper. Instead,
we will rely on the results from [8] that use $f$. These will be treated as
black boxes.

######  Definition 2.

Given a continuous function $g:[0,+\infty)\rightarrow\mathbb{R}$ and
$\gamma,t\in\mathbb{R}$, we define

$\Gamma^{+}_{\gamma}(g,t)=\min\\{x:\gamma x+g(x)\geq t\\},\hskip
22.76228pt\Gamma^{-}_{\gamma}(g,t)=\min\\{x:\gamma x-g(x)\geq t\\},$

where the minimum of the empty set is taken as $+\infty$. We define the
function $h(\gamma)$ as the infimum, over all 1-Lipschitz functions $g$ with
$g(0)=0$, of

$\limsup\limits_{t\rightarrow\infty}\frac{\Gamma^{+}_{\gamma}(g,t)+\Gamma^{-}_{\gamma}(g,t)}{t}.$

We take $f:(0,+\infty)\rightarrow\mathbb{R}$ as

$f(\lambda)=1-\frac{1}{\frac{2\lambda}{(1+\lambda)^{2}}h\left(\frac{\lambda-1}{\lambda+1}\right)+\frac{2\lambda}{1+\lambda}}.$

We extend this definition by continuity to $f(0)=1$ and $f(+\infty)=1/2$.

Figure 1: Plot of the bounds on the function $f(x)$. The conjectured value is
given in blue.

This function satisfies the following bounds:

$\frac{x+1}{2x+1}\leq
f(x)\leq\left\\{\begin{array}[]{ccc}\frac{2x^{2}+3x+7+2\sqrt{x+1}}{4x^{2}+4x+9}&\text{for}&0\leq
x<3,\\\ &&\\\ \frac{x+1}{2x}&\text{for}&x\geq 3,\end{array}\right.$ (1)

and the upper bound is known to be sharp for $x\in[0,1]$. A conjecture in [8]
states that the upper bound is the correct value for all $x$. Note that, with
this definition, $\rho(P_{\infty})=f(1)$. The three results from [8] that
apply to infinite factors, one upper bound and two lower bounds, are as
follows:

######  Theorem 1 ([8]).

  1. (i)

For every non-empty finite graph $F$, we have

$\rho(\omega\cdot F)\leq f\left(\min\limits_{\begin{subarray}{c}I\text{ indep.
in }F\\\ I\neq\emptyset\end{subarray}}\frac{|N(I)|}{|I|}\right).$

  2. (ii)

Let $F$ be a graph, and let $I\subseteq V(F)$ be a non-empty independent set
such that $N(I)$ is also independent. Then $\rho(\omega\cdot F)\geq
f\left(\frac{|N(I)|}{|I|}\right)$.

  3. (iii)

Let $F$ be a finite graph. Then $\rho(\omega\cdot
F)\geq\frac{|V(F)|}{2|V(F)|-\alpha(F)}$.

For certain choices of $F$, these bounds are sufficient to find the exact
value of the Ramsey density. Indeed, if one of the independent sets $I$ that
minimize the ratio $\frac{|N(I)|}{|I|}$ also satisfies that $N(I)$ is
independent, then (i) and (ii) combined give that $\rho(\omega\cdot
F)=\frac{|N(I)|}{|I|}$. This is the case, for example, for all bipartite
graphs.

The bound (iii) is derived from a result of Burr, Erdős and Spencer [1], and
does not depend on the function $f$. There is currently no known graph $F$ for
which this bound is tight and, in fact, in this paper we will show that,
unless the lower bound of (1) is tight for some $x$, the bound (iii) is never
tight.

Two interesting cases in which the exact value does not follow from these
bounds are cliques and odd cycles.

  * •

In the case of cliques, we have $\rho(\omega\cdot K_{n})\leq
f(n-1)\leq\frac{n}{2n-2}$ from (i) and $\rho(\omega\cdot
K_{n})\geq\frac{n}{2n-1}$ from (iii). We cannot obtain any lower bound from
(ii) because there is no non-empty independent set $I$ such that $N(I)$ is
independent, if $n\geq 3.$

  * •

In the case of odd cycles, we have $\rho(\omega\cdot C_{2n+1})\leq
f\left(\frac{n+1}{n}\right)$ from (i), $f(\omega\cdot C_{2n+1})\geq
f\left(\frac{n}{n-1}\right)$ from (ii) (if $n\geq 2$), and $\rho(\omega\cdot
C_{2n+1})\geq\frac{2n+1}{3n+2}$ from (iii). For $n$ large enough, the lower
bound from (ii) is better than that from (iii), although the latter has the
advantage of being easier to compute.

In this paper we will prove the following new lower bound for infinite
factors:

######  Theorem 2.

Let $F$ be a finite graph. Then $\rho(\omega\cdot F)\geq
f\left(\frac{|V(F)|}{\alpha(F)}-1\right)$.

This bound is not weaker than Theorem 1(iii). That result can be expressed as
$\rho(\omega\cdot F)\geq\bar{f}\left(\frac{|V(F)|}{\alpha(F)}-1\right)$, for
$\bar{f}(x)=\frac{x+1}{2x+1}$. By (1), we have $f(x)\geq\bar{f}(x)$.

We can compare this lower bound to the upper bound in Theorem 1(i) to find a
new family of graphs for which we can obtain an exact result:

######  Corollary 1.

Let $F$ be a finite graph. Suppose that, among the independent sets $I$ that
minimize $\frac{|N(I)|}{|I|}$, there is at least one with size $\alpha(F)$.
Then $\rho(\omega\cdot F)=f\left(\frac{|V(F)|}{\alpha(F)}-1\right)$. In
particular:

  * •

$\rho(\omega\cdot K_{n})=f(n-1)$ for all $n\geq 1$.

  * •

$\rho(\omega\cdot C_{2n+1})=f\left(\frac{n+1}{n}\right)$ for all $n\geq 1$.

Since a triangle is both a clique and an odd cycle, these results show that
$\rho(\omega\cdot K_{3})=f(2)$, answering a question from [8]. Note however
that, since we do not know the actual value of $f(2)$, the explicit bounds on
$\rho(\omega\cdot K_{3})$ have not improved: all we know is that
$\frac{3}{5}\leq\rho(\omega\cdot K_{3})\leq(21+\sqrt{12})/33=0.74133\dots$ (we
suspect that the upper bound is sharp). The next result improves upon this
lower bound:

######  Theorem 3.

$\rho(\omega\cdot K_{3})\geq 1-\frac{1}{\sqrt{7}}\approx 0.62204$ (and
therefore $f(2)\geq 1-\frac{1}{\sqrt{7}}$).

Unlike Theorem 2, whose proof is based on the techniques from [2] and [8]
(which is where the function $f$ comes from), Theorem 3 is based on a careful
analysis of the technique of Burr, Erdős and Spencer [1].

##  2 Proof of Theorem 2

The proof of Lemma 2 uses the same overarching outline as Theorem 1(ii): given
an edge coloring of $K_{\mathbb{N}}$ we color the vertices of the graph while
encoding some information, we find a smaller substructure in the resulting
coloring and finally use the information in the vertices to extend this
substructure into our desired graph.

The main difference comes from the choice of coloring. While the proof from
[8] uses a coloring due to Elekes, Soukup, Soukup and Szentmiklóssy [5], here
instead we will use a coloring that arises from a straightforward application
of Ramsey’s theorem. The reason for this change is that the coloring from [5]
is good at finding “connections at infinity”, allowing one to add infinitely
many vertices to a component with little restriction as to how they are
joined. Because the components of $\omega\cdot F$ all have bounded size, we do
not need to worry about connecting more and more vertices into a component.
Instead, we will be able to create each component of $\omega\cdot F$ in one or
two steps.

The intermediate step of finding a smaller substructure is essentially the
same. We will use the following lemma:

######  Lemma 1.

Let $r$ and $s$ be positive integers. For every $\epsilon>0$ there exist
$\tau>0$ and $N$ with the following property: for every graph $G$ on vertex
set $[n]$, with $n>N$, such that $\delta(G)>(1-\tau)n$, in every coloring
$\Psi:V(G)\cup E(G)\rightarrow\\{R,B\\}$ there exists a subgraph $F\subseteq
G$, a color $C\in\\{R,B\\}$ and a value $k\in[\delta n,n]$ such that
$|V(F)\cap[k]|\geq(f(s/r)-\epsilon)k$ and every component of $F$ is of one of
the following types:

  * •

An isolated vertex of color $C$.

  * •

A copy of $K_{r,s}$ in which the edges and the side of size $s$ have color
$C$, and the side of size $r$ has the opposite color.

This is a finitary version of Lemma 2 in [8]. The same proof that was used in
that paper works here.

The next lemma will require the following definition, which will be the key
information that will be encoded into our vertex coloring:

######  Definition 3.

Given a set $X$ of vertices in an edge-colored graph, a real number
$\epsilon>0$, a natural number $s$ and a color $C\in\\{R,B\\}$, we say that
$X$ is $(C,\epsilon,s)$-adequate if every subset of $X$ of size at least
$\epsilon|X|$ contains a $C$-colored clique of size $s$.

Observe that in particular, if a set $X$ is $(C,\epsilon,s)$-adequate, then it
contains a disjoint family of $C$-colored copies of $K_{s}$ covering at least
$(1-\epsilon)|X|$ vertices. This is proved by considering a maximal family of
such cliques.

######  Lemma 2.

For every $\epsilon>0$ and $s\in\mathbb{N}$ there exist $T,N\in\mathbb{N}$
with the following property: for every $\Psi:E(K_{n})\rightarrow\\{R,B\\}$
with $n>N$ there exists a partition $V(K_{n})=V_{1}\cup V_{2}\cup\dots\cup
V_{T}$ into almost equal parts, and colors $C_{1},\dots,C_{T}\in\\{R,B\\}$ in
which all but $\epsilon T$ sets $V_{i}$ are $(C_{i},\epsilon,s)$-adequate.

###### Proof.

Let $\alpha=\lceil 2\epsilon^{-1}\rceil s$ and $q=R(K_{\alpha},K_{\alpha})$.
Take a maximal family $\mathcal{F}$ of disjoint monochromatic copies of
$K_{\alpha}$ ($\mathcal{F}$ might contain cliques of different colors). The
vertex set $V(K_{n})\setminus V(\mathcal{F})$ does not contain any
monochromatic $K_{\alpha}$, so its size is at most $q$.

Let $T=\lceil 3\epsilon^{-1}\rceil$. Each of the sets $V_{1},\dots,V_{T}$ will
have size either $\lfloor n/T\rfloor$ or $\lceil n/T\rceil$. Let
$\beta=\lfloor\lfloor n/T\rfloor/\alpha\rfloor$, this is the number of
$\alpha$-cliques that fit into a set $V_{i}$ (of the smaller type). Let
$\mathcal{F}_{R}$ and $\mathcal{F}_{B}$ be the sets of red and blue cliques
from $\mathcal{F}$, respectively.

Create $V_{1},V_{2},\dots,V_{T}$, all initially empty. For
$\lfloor|\mathcal{F}_{R}|/\beta\rfloor+\lfloor|\mathcal{F}_{B}|/\beta\rfloor$
of these sets, put $\beta$ cliques from $\mathcal{F}$ of the same color into
each. We call these sets pseudo-adequate, and we will later show that they are
indeed adequate. Distribute the remaining vertices from $K_{n}$ into the sets
$V_{i}$ so that the resulting sets are almost equal. The number of pseudo-
adequate sets is

$\left\lfloor\frac{|\mathcal{F}_{R}|}{\beta}\right\rfloor+\left\lfloor\frac{|\mathcal{F}_{B}|}{\beta}\right\rfloor\geq\left\lfloor\frac{|\mathcal{F}|}{\beta}\right\rfloor-1\geq\frac{\mathcal{F}}{\beta}-2\geq\frac{\frac{n-q}{\alpha}}{\beta}-2\geq
T-\frac{q}{\alpha\beta}-2\geq T-3$

if $\beta\geq q$ (this will be the case if $n>q\alpha T$). That means that
there are at most three sets which are not pseudo-adequate, which represents
less than $\epsilon T$ of the sets.

For each pseudo-adequate set $V_{i}$, we let $C_{i}$ be the color of the
cliques from $\mathcal{F}$ that were put into it. For the (up to three)
remaining sets $V_{i}$, choose $C_{i}$ arbitrarily. If $V_{i}$ is pseudo-
adequate, there are at most $\alpha$ vertices not in a copy of $K_{\alpha}$
from $\mathcal{F}$.

In a pseudo-adequate set $V_{i}$, take a subset $S\subseteq V_{i}$ of size at
least $\epsilon|V_{i}|$. There are at least $\epsilon|V_{i}|-\alpha$ vertices
in $S$ which belong to a clique in $\mathcal{F}$, so by the pigeonhole
principle $S$ contains at least
$\lceil\frac{\epsilon|V_{i}|-\alpha}{\beta}\rceil$ elements from some clique
in $\mathcal{F}$. This is at least

$\frac{\epsilon|V_{i}|-\alpha}{\beta}\geq\frac{\epsilon\lfloor\frac{n}{T}\rfloor-\alpha}{\left\lfloor\frac{\left\lfloor\frac{n}{T}\right\rfloor}{\alpha}\right\rfloor}\geq\frac{\frac{\epsilon}{2}\lfloor\frac{n}{T}\rfloor}{\frac{\left\lfloor\frac{n}{T}\right\rfloor}{\alpha}}=\frac{\alpha\epsilon}{2}\geq
s$

if $\alpha<\frac{\epsilon}{2}\left\lfloor\frac{n}{T}\right\rfloor$. This holds
if $n>\lceil 2\epsilon^{-1}\alpha\rceil T$. We conclude that we can take
$N=\max\\{q\alpha T,\lceil 2\epsilon^{-1}\alpha\rceil T\\}$.∎

We will prove a finitary version of Theorem 2, from which the infinite version
follows directly:

######  Lemma 3.

For every finite graph $F$ and every $\epsilon>0$ there exists $N$ and
$\tau>0$ with the following property: for every $n>N$, for every coloring
$\Psi:E(K_{n})\rightarrow\\{R,B\\}$ there exists $t\in[\tau n,n]$ and a
monochromatic family $\mathcal{F}$ of disjoint copies of $F$ such that

$\frac{|V(\mathcal{F})\cap[t]|}{t}\geq
f\left(\frac{|V(F)|}{\alpha(F)}-1\right)-\epsilon.$

###### Proof of Theorem 2.

Fix $\epsilon>0$. Let $\Psi:E(K_{\mathbb{N}})\rightarrow\\{R,B\\}$ be an edge-
coloring. Consider a sequence $n_{1}<n_{2}<\dots$ of natural numbers with
$\lim n_{i+1}/n_{i}\rightarrow\infty$. For each $i\geq 2$, consider the
colored clique $G_{i}$ on vertex set $(n_{i-1},n_{i}]$, whose edge-coloring is
$\Psi_{i}$, the one induced by $\Psi$. Now consider the graph
$G_{i}^{\prime}$, obtained by substracting $n_{i-1}$ from the label on each
vertex.

By Lemma 3, there exists a fixed $\tau>0$ such that, for every $i$ large
enough (enough to have $n_{i}-n_{i-1}>N$) there is a value
$t_{i}\in[\tau(n_{i}-n_{i-1}),n_{i}-n_{i-1}]$, and a monochromatic family
$\mathcal{F}^{\prime}_{i}$ of disjoint copies of $F$ in $G^{\prime}_{i}$ with
color $C_{i}$, such that

$\frac{|V(\mathcal{F}^{\prime}_{i})\cap[t_{i}]|}{t_{i}}\geq
f\left(\frac{|V(F)|}{\alpha(F)}-1\right)-\epsilon.$

For infinitely many values of $i$, the color $C_{i}$ will be the same, which
we denote $C$. Let $\mathcal{F}_{i}$ be family of copies of $F$ in
$K_{\mathbb{N}}$ obtained by adding $n_{i-1}$ to every vertex in
$\mathcal{F}^{\prime}_{i}$. Then consider
$\mathcal{F}=\cup_{C_{i}=C}\mathcal{F}_{i}$. Clearly $\mathcal{F}$ is a
monochromatic copy of $\omega\cdot F$. We will show that it has upper density
at least $f\left(|V(F)|/\alpha(F)-1\right)-\epsilon$.

For every $i$ with $C_{i}=C$, we have

$\displaystyle\frac{|V(\mathcal{F})\cap[t_{i}+n_{i-1}]|}{t_{i}+n_{i-1}}\geq$
$\displaystyle\frac{|V(\mathcal{F}^{\prime}_{i})\cap[t_{i}]|}{t_{i}+n_{i-1}}\geq\left(f\left(\frac{|V(F)|}{\alpha(F)}-1\right)-\epsilon\right)\frac{t_{i}}{t_{i}+n_{i-1}}$
$\displaystyle\geq$
$\displaystyle\left(f\left(\frac{|V(F)|}{\alpha(F)}-1\right)-\epsilon\right)\frac{\tau
n_{i}}{\tau n_{i}+n_{i-1}}.$

By taking the upper limit on the latter expression as $i\rightarrow\infty$, we
conclude that $\bar{d}(\mathcal{F})\geq
f\left(|V(F)|/\alpha(F)-1\right)-\epsilon$, and thus this value is a lower
bound on $\rho(\omega\cdot F)$. Since this is valid for all $\epsilon>0$, we
conclude $\rho(\omega\cdot F)\geq f(|V(F)|/\alpha(F)-1)$.∎

In the proof of Lemma 3 we routinely omit rounding signs, except when they are
part of a definition. All rounding errors are smaller than the effect that
taking stronger constants.

###### Proof of Lemma 3.

Let $\gamma=\gamma(\epsilon,F)$ be small enough (we will define how small
later). Let $\ell=\lceil\gamma^{-1}\rceil$. Partition $[n]$ into $\ell$ almost
equal intervals, labelled $I_{1},I_{2},\dots,I_{\ell}$ from smallest to
largest. Apply the colored variant of the Szemerédi regularity lemma [7] to
this edge-colored graph $K_{n}$ to find $M=M(\gamma)$ and a $\gamma$-regular
partition $\\{X_{0},X_{1},X_{2},\dots,X_{m}\\}$, with $m\leq M$, that refines
$\\{I_{1},\dots,I_{\ell}\\}$.

Now let $\kappa=\kappa(\epsilon,F)$ be small enough ($\kappa$ will be chosen
before $\gamma$, so we may suppose $\gamma\ll\kappa\ll\epsilon$). Lemma 2
gives $T,N^{\prime}(\kappa,s^{\prime})$, for $s^{\prime}=|V(F)|$. If the size
of all $X_{i}$ with $i>0$ is at least $N^{\prime}$ (which will happen if
$n>(1-\gamma)^{-1}mN^{\prime}$, and in particular is implied by
$N>(1-\gamma)^{-1}MN^{\prime}$) then by Lemma 2 we can subdivide each $X_{i}$
into $T$ parts $X^{i}_{1},\dots,X^{i}_{T}$, as in the statement of that lemma.
Because the sizes of the $X_{i}$ are all almost equal, the $Tm$ sets
$X^{i}_{j}$ also have almost equal sizes. Color each set $X^{i}_{j}$ with the
color $C^{i}_{j}$ that it receives from Lemma 2. Assuming that they are all
non-empty (true if $N>(1-\gamma)^{-1}TM$) we can label them
$Y_{1},\dots,Y_{Tm}$, with the property that $\min Y_{1}<\min
Y_{2}<\ldots<\min Y_{Tm}$. We denote the color of $Y_{i}$ by $C_{i}$.

If the pair $(X_{i_{1}},X_{i_{2}})$ is $\gamma$-regular, then the pair
$(X^{i_{1}}_{j_{1}},X^{i_{2}}_{j_{2}})$ is $T\gamma$-regular. Construct an
auxiliary colored graph $J$ on $[Tm]$ as follows: the color of a vertex $v$ is
the color of $Y_{v}$. If the pair $(Y_{v},Y_{w})$ is $T\gamma$-regular in the
original graph, draw an edge $vw$ in $J$, whose color is the densest color in
the bipartite graph $(Y_{i},Y_{j})$ (ties are broken arbitrarily). Because
every $X_{i}$ is $\gamma$-regular with at least $(1-\gamma)m$ other $X_{j}$,
each $Y_{i}$ is $T\gamma$-regular with at least $(1-\gamma)Tm$ other $Y_{j}$.
The minimum degree of $J$ is at least $(1-\gamma)Tm$.

Next we apply Lemma 1 to this totally colored graph. This produces a value
$k\in[\delta Tm,Tm]$, a color $C$ and a subgraph $J^{\prime}\subseteq J$ in
which every component is either an isolated vertex of color $C$ or a
$K_{r,s}$, for $r=\alpha(F)$ and $s=|V(F)|-\alpha(F)$, colored as in the
statement of Lemma 1. These satisfy

$\frac{|V(J^{\prime})\cap[k]|}{k}\geq
f\left(\frac{s}{r}\right)-\frac{\epsilon}{2}.$

This assumes that the minimum degree condition is satisfied, which is true if
$\gamma\ll\tau=\tau(\epsilon,F)$. We also have $\delta=\delta(\epsilon,F)$,
with these two functions as in Lemma 1.

Now we return to our original graph on $[n]$ with a coloring given by $\Psi$.
We will find in it a family $\mathcal{F}$ of vertex-disjoint copies of $F$,
such that every copy is contained in parts $Y_{i}$ corresponding to a
component of $J^{\prime}$, according to the following restrictions:

  * •

If the corresponding component of $J^{\prime}$ is an isolated vertex $v$, then
there is no restriction: any copy of $F$ in $Y_{v}$ can be taken.

  * •

If the corresponding component of $J^{\prime}$ is a $K_{r,s}$ on vertex
classes $\\{v_{1},\dots,v_{r}\\}$ and
$\\{v^{\prime}_{1},\dots,v^{\prime}_{s}\\}$, we only consider the copies of
$F$ which contain exactly $r$ vertices in $Y_{v_{1}}\cup\dots\cup Y_{v_{r}}$,
and $s$ vertices in $Y_{v^{\prime}_{1}}\cup\dots\cup Y_{v^{\prime}_{s}}$.

Under these restrictions we consider an inclusion-maximal such family
$\mathcal{F}$. Let $t=\min Y_{k}$. We will show that
$\frac{|V(\mathcal{F})\cap[t]|}{t}$ is large enough for our purposes.

When each set $X_{i}$ is split into $T$ sets $X_{i}^{1},\dots,X_{i}^{T}$,
there are at most $\kappa T$ of them which cannot be almost-partitioned into
cliques $K_{s^{\prime}}$ of the corresponding color. In these sets
$X_{i}^{j}$, assuming they are isolated vertices in $J$, the family
$\mathcal{F}$ will use at least $(1-\kappa)|X_{i}^{j}|$ vertices in each of
them.

On the other hand, consider a component in $J$ which is a copy of $K_{r,s}$ on
vertex classes $\\{v_{1},\dots,v_{r}\\}$ and
$\\{v^{\prime}_{1},\dots,v^{\prime}_{s}\\}$. There are at most $\kappa Tm$ of
these components which contain a set which is not adequate, in the sense of
Lemma 2. We claim that, if all of the sets are adequate, then $\mathcal{F}$
contains at least a $1-\xi$ proportion of the vertices in
$Y_{v_{1}}\cup\dots\cup Y_{v_{r}}\cup Y_{v^{\prime}_{1}}\cup\dots\cup
Y_{v^{\prime}_{s}}$, for some $\xi$ that will be defined later. Indeed,
suppose that fewer than a $1-\xi$ proportion of vertices is contained in
$\mathcal{F}$. Then there are two sets, w.l.o.g. $Y_{v_{1}}$ and
$Y_{v^{\prime}_{1}}$, such that $\frac{|Y_{v_{1}}\cap
V(\mathcal{F})|}{|Y_{v_{1}}|},\frac{|Y_{v^{\prime}_{1}}\cap
V(\mathcal{F})|}{|Y_{v^{\prime}_{1}}|}<1-\xi$. Let $W=Y_{v_{1}}\setminus
V(\mathcal{F})$ and $W^{\prime}=Y_{v^{\prime}_{1}}\setminus V(\mathcal{F})$.
Because the pair $(Y_{v_{1}},Y_{v^{\prime}_{1}})$ is $T\gamma$-regular, and
has density at least $\frac{1}{2}$ in color $C$, the bipartite graph
$(W,W^{\prime})$ has density at least $\frac{1}{2}-T\gamma>\frac{1}{4}$ in
color $C$.

Next we will select $w_{1},w_{2},\dots,w_{r}$, which will be the vertices of
our copy of $F$ in $Y_{v_{1}}$. Select $w_{1}\in W$ that maximizes the number
of $C$-neighbors in $W^{\prime}$. Then let $w_{2}\in W\setminus\\{w_{1}\\}$
that maximizes the size of its $C$-neighborhood in the $C$-neighborhood of
$w_{1}$ in $W^{\prime}$. We proceed this way, each time picking the vertex
that maximizes the common neighborhood with the previous choices. As long as
$\xi 4^{-r}>T\gamma$ (which will be true because $\gamma\ll\xi,T^{-1}$), we
can use regularity to ensure that the density of the $C$-colored edges between
the common neighborhood and $W$ is at least $\frac{1}{4}$.

Let $Z$ be the common $C$-neighborhood of $w_{1},w_{2},\dots,w_{r}$ in
$W^{\prime}$. It has size at least $4^{-r}|W^{\prime}|\geq\xi
4^{-r}|Y_{v^{\prime}_{1}}|$. If $\xi 4^{-r}>\kappa$ (which again holds because
$\kappa\ll\xi)$, by the partition of Lemma 2 there is a clique of size
$s^{\prime}=r+s$ of color $C$ in $Z$. We can take
$w^{\prime}_{1},w^{\prime}_{2},\dots,w^{\prime}_{s}$ in this clique as the
remaining vertices of $F$. We have thus found a new copy of $F$, which
contradicts the maximality of $\mathcal{F}$. This completes the proof that
$\mathcal{F}$ contains at least a $1-\xi$ proportion of the vertices in
$Y_{v_{1}}\cup\dots\cup Y_{v_{r}}\cup Y_{v^{\prime}_{1}}\cup\dots\cup
Y_{v^{\prime}_{s}}$.

We are ready to estimate
$\frac{|V(\mathcal{F})\cap[t]|}{t}=1-\frac{|[t]\setminus V(\mathcal{F})|}{t}$.
We will do so by giving a lower bound on $t$, and an upper bound on the
vertices of $[t]$ which do not belong to $\mathcal{F}$.

Since each set $Y_{i}$ satisfies $\max Y_{i}-\min
Y_{i}\leq\ell^{-1}n\leq\gamma n$ (because they are contained in some interval
$I_{j}$), from $t=\min Y_{k}$ we have $Y_{1}\cup\dots\cup
Y_{k}\subseteq[t+\gamma n]$. It is also true that $|Y_{1}\cup\dots\cup
Y_{Tm}|=|[n]\setminus X_{0}|\geq(1-\gamma)n$. Since the $|Y_{i}|$ are almost
equal, we have $|Y_{i}|\geq\frac{(1-\gamma)n}{Tm}$, and

$t=|[t+\gamma n]|-\gamma n\geq k|Y_{i}|-\gamma
n\geq\frac{(1-\gamma)n}{Tm}k-\gamma n.$

Next we will estimate $|[t]\setminus V(\mathcal{F})|$. Every vertex $z$ which
belongs to $[t]$ but not to $V(\mathcal{F})$ is in at least one of the
following classes:

  * •

They are in $X_{0}$.

  * •

They are in a class $Y_{v}$ with $v\notin J^{\prime}$.

  * •

They are in a class $Y_{v}$, where $v$ is an isolated vertex in $J^{\prime}$,
but $Y_{v}$ is not adequate as in Lemma 2.

  * •

They are in a class $Y_{v}$, where $v$ is an isolated vertex in $J^{\prime}$,
and $Y_{v}$ is adequate, but $z$ is not in $\mathcal{F}$.

  * •

They are in a class $Y_{v}$ with $v$ belonging to a $K_{r,s}$ in $J^{\prime}$,
where some sets $Y_{v^{\prime}}$ corresponding to a vertex of this component
is not adequate.

  * •

They are in a class $Y_{v}$ with $v$ belonging to a $K_{r,s}$ in $J^{\prime}$,
and all sets $Y_{v^{\prime}}$ corresponding to vertices of this component are
adequate, but $z$ is not in $\mathcal{F}$.

The sum of the number of vertices in each class can be upper-bounded by

$\gamma
n+\left(1-f\left(\frac{s}{r}\right)+\frac{\epsilon}{2}\right)k\frac{n}{Tm}+\kappa
n+\kappa n+(r+s)\kappa n+\xi n,$

respectively. If $\xi,\kappa$ and $\gamma$ are chosen small enough after
choosing $\epsilon$, this can be upper-bounded by

$|[t]\setminus
V(\mathcal{F})|\leq\left(1-f\left(\frac{s}{r}\right)+\frac{2\epsilon}{3}\right)k\frac{n}{Tm}.$

Combining both bounds, we have

$\displaystyle\frac{|V(\mathcal{F})\cap[t]|}{t}=$ $\displaystyle
1-\frac{|[t]\setminus V(\mathcal{F})|}{t}\geq
1-\frac{\left(1-f\left(\frac{s}{r}\right)+\frac{2\epsilon}{3}\right)k\frac{n}{Tm}}{\frac{(1-\gamma)n}{Tm}k-\gamma
n}$ $\displaystyle=$ $\displaystyle
1-\frac{1-f\left(\frac{s}{r}\right)+\frac{2\epsilon}{3}}{1-\gamma-\gamma\frac{Tm}{k}}\geq
1-\frac{1-f\left(\frac{s}{r}\right)+\frac{2\epsilon}{3}}{1-\gamma-\gamma\delta^{-1}}\geq
f\left(\frac{s}{r}\right)-\epsilon,$

since $\gamma\ll\delta,\epsilon$. To complete the proof of Lemma 3, we observe
that there exists $\tau>0$ such that $t\geq\tau n$, namely

$t\geq\frac{(1-\gamma)n}{Tm}k-\gamma
n\geq\left(\delta(1-\gamma)-\gamma\right)n.\qed$

##  3 Proof of Theorem 3

The proof of Theorem 3 borrows ideas from the proof by Burr, Erdős and Spencer
[1] for the Ramsey number of $n$ disjoint triangles, $R(n\cdot K_{3},n\cdot
K_{3})=5n$. The following configuration plays a crucial role in both results:

######  Definition 4.

A bowtie is an edge-colored graph on five vertices, formed by a red triangle
and a blue triangle sharing a vertex.

######  Lemma 4.

Let $\Psi:K_{n}\rightarrow\\{R,B\\}$, let $F$ be the largest monochromatic
family of disjoint triangles, and let $F^{\prime}$ be the largest family of
disjoint bowties. Then $3|F|+2|F^{\prime}|\geq n-5$.

###### Proof.

We start by noting that, if a two-edge-colored $K_{6}$ contains a vertex-
disjoint red triangle and a blue triangle, then it also contains a bowtie.
This is because out of the nine edges between both triangles, there are at
least five with the same color, w.l.o.g. red. By the pigeonhole principle,
there is a vertex $v$ in the blue triangle incident to at least two red edges.
Then the original blue triangle plus the two red-neighbors of $v$ form a
bowtie.

Now consider the largest family $F^{\prime}$ of vertex-disjoint bowties. The
set of vertices not in $F^{\prime}$ cannot contain triangles in both colors,
by the argument above. Suppose w.l.o.g. that all remaining triangles are red.
Let $F^{\prime\prime}$ be a maximal family of vertex-disjoint red triangles in
$V(K_{n})\setminus V(F^{\prime})$. We know that
$|V(F^{\prime})|+|V(F^{\prime\prime})|\geq n-5$, because the remaining
vertices do not contain a monochromatic triangle in either color. Now note
that there is a family of disjoint red triangles in $K_{n}$ of size
$|F^{\prime}|+|F^{\prime\prime}|$, obtained by taking the triangles in
$F^{\prime\prime}$ and the red triangles in each bowtie of $F^{\prime}$. Thus
we get $|F|\geq|F^{\prime}|+|F^{\prime\prime}|$, and

$3|F|+2|F^{\prime}|\geq
3(|F^{\prime}|+|F^{\prime\prime}|)+2|F^{\prime}|=5|F^{\prime}|+3|F^{\prime\prime}|=|V(F^{\prime})|+|V(F^{\prime\prime})|\geq
n-5,$

as we wanted to prove.∎

The proof of the following lemma is a simple (but slightly cumbersome) case
analysis:

######  Lemma 5.

Let $W_{1},W_{2}$ be two vertex-disjoint bowties in a red-blue edge-coloring
of $K_{10}$. Then either there are two vertex-disjoint triangles of the same
color using four vertices of $W_{1}$ or there are nine vertices containing two
vertex-disjoint red triangles and two vertex-disjoint blue triangles.

Figure 2: Labeling of the vertices of $W_{1}$ and $W_{2}$.

###### Proof.

We label the vertices of $W_{1}$ and $W_{2}$ as in Figure 2. Whenever a target
configuration from the statement appears, we will denote it by two or four
triangles between square brackets. We suppose that the coloring is such that
none of the configurations from the statement appears, and we will reach a
contradiction.

Without loss of generality the edge $C_{1}C_{2}$ is blue. At least one of the
edges $B_{2}R_{3}$ or $B_{2}R_{4}$ must be blue, because otherwise
$[C_{1}R_{1}R_{2},B_{2}R_{3}R_{4}]$. We will assume w.l.o.g. that $B_{2}R_{3}$
is blue. For the same reason, we assume that $R_{2}B_{3}$ is red. Then the
color of some edges is forced:

  * •

$C_{1}R_{3}$ is red, otherwise
$[C_{1}R_{1}R_{2},C_{2}R_{3}R_{4},C_{1}B_{2}R_{3},C_{2}B_{3}B_{4}]$.

  * •

$C_{1}B_{3}$ is blue, otherwise
$[C_{1}R_{2}B_{3},C_{2}R_{3}R_{4},C_{1}B_{1}B_{2},C_{2}B_{3}B_{4}]$.

  * •

$B_{1}R_{3}$ is red, otherwise
$[C_{1}R_{1}R_{2},C_{2}R_{3}R_{4},B_{1}B_{2}R_{3},C_{1}C_{2}B_{3}]$.

  * •

$B_{1}C_{2}$ is blue, otherwise $[B_{1}C_{2}R_{3},C_{1}R_{1}R_{2}]$.

  * •

$B_{1}R_{4}$ is blue, otherwise $[B_{1}R_{3}R_{4},C_{1}R_{1}R_{2}]$.

  * •

$B_{2}R_{4}$ is red, otherwise
$[C_{1}R_{1}R_{2},C_{2}R_{3}R_{4},B_{1}B_{2}R_{4},C_{1}C_{2}B_{3}]$.

  * •

$B_{2}C_{2}$ is blue, otherwise $[C_{1}R_{1}R_{2},B_{2}C_{2}R_{4}]$.

  * •

$C_{1}R_{4}$ is red, otherwise
$[C_{1}R_{1}R_{2},C_{2}R_{3}R_{4},C_{1}B_{1}R_{4},C_{2}B_{3}B_{4}]$.

At this point, we split our analysis into two cases, depending on the color of
$R_{1}B_{1}$. If $R_{1}B_{1}$ is blue:

  * •

$R_{1}B_{2}$ is red, otherwise $[B_{1}B_{2}R_{1},C_{1}C_{2}B_{3}]$.

  * •

$R_{1}R_{4}$ is red, otherwise $[B_{1}R_{1}R_{4},C_{1}C_{2}B_{2}]$.

  * •

$R_{2}R_{3}$ is blue, otherwise $[B_{2}R_{1}R_{4},C_{1}R_{2}R_{3}]$.

But then we reach a contradiction with the color of $R_{2}B_{2}$. If it is
red, then $[R_{1}R_{2}B_{2},C_{1}R_{3}R_{4}]$. If it is blue, then
$[R_{2}B_{2}R_{3},C_{1}C_{2}B_{1}]$.

We suppose now that $R_{1}B_{1}$ is red. Then:

  * •

$B_{1}R_{2}$ is blue, otherwise $[R_{1}R_{2}B_{1},C_{1}R_{3}R_{4}]$.

  * •

$B_{2}R_{2}$ is red, otherwise $[B_{1}B_{2}R_{2},C_{1}C_{2}B_{3}]$.

  * •

$R_{2}R_{4}$ is red, otherwise $[B_{1}R_{2}R_{4},C_{1}C_{2}B_{2}]$.

  * •

$R_{1}R_{3}$ is blue, otherwise $[B_{1}R_{1}R_{3},C_{1}R_{2}R_{4}]$.

But then we reach a contradiction with the color of $R_{1}B_{2}$. If it is
red, then $[R_{1}R_{2}B_{2},C_{1}R_{3}R_{4}]$. If it is blue, then
$[R_{1}R_{3}B_{2},B_{1}C_{1}C_{2}]$. This concludes the case analysis.∎

We now have the necessary preparations for the proof of Theorem 3. Similarly
to the proof of Theorem 2, we will here prove a finitary version. The reason
that Theorem 3 follows from the lemma below is the same argument by which
Theorem 2 follows from Lemma 3.

######  Lemma 6.

Let $\delta=\frac{4\sqrt{7}+2}{27}=0.46603\dots$ and
$\gamma=1-\frac{1}{\sqrt{7}}=0.62203\dots$. For every
$\Psi:E(K_{n})\rightarrow\\{R,B\\}$ there exists $k\in\\{\lfloor\delta
n\rfloor,n\\}$ and a monochromatic family of disjoint triangles $F$ with
$|V(F)\cap[k]|\geq\gamma k-7$.

###### Proof.

Let $F_{1}$ be a maximum monochromatic family of disjoint triangles, and
$F^{\prime}_{1}$ be a maximum family of disjoint bowties, both in $[\delta
n]$. Let $F_{2},F^{\prime}_{2}$ be the families defined similarly in $(\delta
n,n]$. If $3|F_{1}|\geq\gamma\delta n-7$, or
$3|F_{1}|+3|F^{\prime}_{2}|\geq\gamma n-7$, or
$3|F^{\prime}_{1}|+3|F_{2}|\geq\gamma n-7$, then we are done. Therefore we
assume the opposite.

Suppose first that $|F^{\prime}_{1}|\geq|F^{\prime}_{2}|$. This leads to a
contradiction, because, by Lemma 4,
$0<(3|F_{1}|+2|F^{\prime}_{1}|-\lfloor\delta
n\rfloor+5)+2(3|F_{2}|+2|F^{\prime}_{2}|-(1-\delta)n+5)+(\gamma\delta
n-7-3|F_{1}|)+2(\gamma
n-7-3|F^{\prime}_{1}|-3|F_{2}|)+4(|F^{\prime}_{1}|-|F^{\prime}_{2}|)<(\gamma\delta+2\gamma+\delta-2)n-5<0$.
Therefore we will assume $|F^{\prime}_{2}|\geq|F^{\prime}_{1}|$.

Let $q=|F^{\prime}_{1}|$. Let
$(W_{1}^{1},W_{2}^{1}),(W_{1}^{2},W_{2}^{2}),\dots,(W_{1}^{q},W_{2}^{q})$ be a
family of pairs of bowties with $W_{1}^{i}\subseteq[\delta n]$ and
$W_{2}^{i}\subseteq(\delta n,n]$. For every pair $(W_{1}^{i},W_{2}^{i})$ there
is either a set of two disjoint triangles of the same color which contain four
vertices of $W_{1}^{i}$ or nine vertices in $V(W_{1}^{i}\cup W_{2}^{i})$ which
contain two disjoint triangles of each color.

Let $q_{1}$ and $q_{2}$ be the number of pairs $(W_{1}^{i},W_{2}^{i})$ for
which we obtain the former or the latter, respectively. We have
$q_{1}+q_{2}\geq q$. We can find a monochromatic family of triangles that uses
$3(q-\frac{q_{1}}{2})+4\frac{q_{1}}{2}=3q+\frac{q_{1}}{2}$ vertices from
$[\delta n]$, and a monochromatic family of triangles that uses
$6q_{2}+\frac{3}{5}(n-9q_{2})=\frac{3}{5}(n+q_{2})$ vertices from $[n]$.
Therefore we must assume $3q+\frac{q_{1}}{2}<\gamma\delta n-7$ and
$\frac{3}{5}(n+q_{2})<\gamma n-7$. But then we have
$0<7(3|F_{1}|+2q-\lfloor\delta n\rfloor+5)+7(\gamma\delta
n-7-3|F_{1}|)+4(\gamma\delta n-7-3q-\frac{q_{1}}{2})+\frac{10}{3}(\gamma
n-7-\frac{3}{5}(n+q_{2}))+2(q_{1}+q_{2}-q)<(11\gamma\delta+\frac{10}{3}\gamma-7\delta-2)n-\frac{175}{3}<0$,
a contradiction.

∎

##  4 Open problems

As we have seen, there are many different conditions that guarantee that the
upper bound on $\rho(\omega\cdot F)$ from Theorem 1(i) is tight. This is the
case if an independent set that minimizes $\frac{|N(I)|}{|I|}$ is maximal (in
which case it matches Theorem 2) or if $N(I)$ is also independent (in which
case it matches Theorem 1(ii)).

Remarkably, we do not know of any graph $F$ for which Theorem 1(i) is not
tight. That leads to the following question:

######  Question 1.

Is it true that, for every finite graph $F$, we have

$\rho(\omega\cdot F)=f\left(\min\limits_{\begin{subarray}{c}I\text{ indep. in
}F\\\ I\neq\emptyset\end{subarray}}\frac{|N(I)|}{|I|}\right)\text{?}$

The smallest graphs $F$ which are not covered by the previous cases have seven
vertices, and three of them are depicted in Figure 3.

Figure 3: Three vertex-minimum graphs $F$ for which $\rho(\omega\cdot F)$ is
not known. If Question 1 has an affirmative answer, their Ramsey densities are
$f(2)$, $f(1)$ and $f(2/3)$, respectively.

In Theorem 3, we improve the lower bound on $f(2)$. There is no reason why the
same argument cannot be used to improve the bound on $f(x)$ for all $x>1$, but
there is a catch: a straightforward generalization can only be applied for
rational $x$, and even then the method appears to be scale-dependent. To
clarify this, consider the graphs $F_{1}=K_{a+b}\setminus K_{b}$ (the graph on
$a+b$ vertices whose complement is an $a$-clique) and
$F_{2}=K_{2(a+b)}\setminus K_{2b}$. They satisfy $\rho(\omega\cdot
F_{1})=f(a/b)=\rho(\omega\cdot F_{2})$ by Corollary 1. However, it is not
clear what the analogous for Lemma 5 should be in each case, and different
versions might produce different bounds on $f(a/b)$, if applied to $F_{1}$ and
$F_{2}$.

######  Question 2.

By adapting the method of Theorem 3, is it possible to find a closed
expression for a continuous function $\bar{f}(x)$ such that
$\frac{x+1}{2x+1}<\bar{f}(x)\leq f(x)$ for all $x>1$?

##  5 Acknowledgements

The research presented here was conducted during a visit by the second author
to the University of Illinois at Urbana-Champaign. We would like to thank the
Berlin Mathematical School for funding this trip.

## References

  * [1] S. A. Burr, P. Erdős, and J. H. Spencer.  Ramsey theorems for multiple copies of graphs.  Trans. Amer. Math. Soc., 209, 1975.
  * [2] J. Corsten, L. DeBiasio, A. Lamaison, and R. Lang.  Upper density of monochromatic infinite paths.  Adv. Comb., 2019.
  * [3] J. Corsten, L. DeBiasio, and P. McKenney.  Density of monochromatic infinite subgraphs ii.  https://arxiv.org/abs/2007.14277, 2020.
  * [4] L. DeBiasio and P. McKenney.  Density of monochromatic infinite subgraphs.  Combinatorica, 39, 2019.
  * [5] M. Elekes, D. T. Soukup, L. Soukup, and Z. Szentmiklóssy.  Decompositions of edge-colored infinite complete graphs into monochromatic paths.  Discrete Math., 340, 2017.
  * [6] P. Erdős and F. Galvin.  Monochromatic infinite paths.  Discrete Math., 113, 1993.
  * [7] J. Komlós and M. Simonovits.  Szemerédi’s Regularity Lemma and its applications in graph theory.  In D. Miklós, V. T. Sós, and T. Szőnyi, editors, Combinatorics, Paul Erdős is Eighty, volume 2, pages 295–352. Bolyai Society Mathematical Studies, 1996.
  * [8] A. Lamaison.  Ramsey upper density of infinite graphs.  https://arxiv.org/abs/2003.06329, 2020.
  * [9] A. Lo, N. Sanhueza-Matamala, and G. Wang.  Density of monochromatic infinite paths.  Electron. J. Combin., 25, 2018.