Stability, dark energy parameterization and swampland aspect of Bianchi
Type-$VI_{h}$ cosmological models with f(R, T)-gravity
Archana Dixit1 and Anirudh Pradhan2

1,2Department of Mathematics, Institute of Applied Sciences and Humanities,
GLA University

Mathura-281 406, Uttar Pradesh, India

1E-mail<EMAIL_ADDRESS>
2E-mail<EMAIL_ADDRESS>
###### Abstract

Stability, dark energy (DE) parameterization and swampland aspects for the
Bianchi form-$VI_{h}$ universe have been formulated in an extended gravity
hypothesis. Here we have assumed a minimally coupled geometry field with a
rescaled function of $f(R,T)$ replaced in the geometric action by the Ricci
scalar $R$. Exact solutions are sought under certain basic conditions for the
related field equations. For the following theoretically valid premises, the
field equations in this scalar-tensor theory have been solved. It is observed
under appropriate conditions that our model shows a decelerating to
accelerating phase transition property. Results are observed to be coherent
with recent observations. Here, our models predict that the universe’s rate of
expansion will increase with the passage of time. The physical and geometric
aspects of the models are discussed in detail. In this model, we also analyze
the parameterizations of dark energy by fitting the EoS parameter $\omega(z)$
with redshift. The results obtained would be useful in clarifying the
relationship between dark energy parameters. In this, we also explore the
correspondence of quintessence dark energy with swampland criteria. The
swampland criteria have been also shown the nature of the scalar field and the
potential of the scalar field.

Keywords: Bianchi type-$VI_{h}$ space-time; $f(R,T)$ gravity; dark energy; EoS
parameterization.

Mathematics Subject Classification 2020: 83C56, 83F05, 83B05

##  1 Introduction

Our view of cosmology has been revolutionized by recent cosmological
observations. This appears that there is an exponential expansion of the
currently observable universe [1]-[3]. The source that drives this
acceleration is termed as ‘dark energy’, whose beginning in present-day
cosmology is still a mystery. This is a result of the way that we do not have,
up until now, a predictable hypothesis of quantum gravity. New prove from
astronomy and cosmology has recently revealed a rather surprising image of the
Universe. Our new datasets from different sources, for example, Cosmic
Microwave Background Radiation(CMBR) and Supernovae, imply that the universe’s
vitality spending plan resembles as: $4\%$ typical baryonic matter, $22\%$
dark matter, and $74\%$ dark energy [4]-[7].
One of the hot topic issues between cosmologists and astrophysicists is the
accelerated problem of expansion of the universe. The common view suggests
that it is dark energy and updated alternative theories to explain the
expansion of the universe. Big alternative theories are $f(R)$ [8] theory,
$f(G)$ [9] theory, $f(R,T)$ [10] theory, and so on. Author [10] proposes the
concept of $f(R,T)$ to describe accelerated universe expansion. They come up
with three models to overcome the argument. They also constructed a remodelled
theory of F(R, T) gravity in which the gravitational Lagrangian is described
by an arbitrary function of The Ricci scalar $R$ and the stress energy tensor
trace $T$. Bianchi type models with anisotropic spatial segments are
fascinating as in they are more broad than the Friedman models. Despite the
fact that there is a solid discussion going on the reasonability of Bianchi
type models [11], these models can be helpful in the portrayal of early
inflationary stage and with appropriate component can be diminished to
isotropic conduct at late occasions. In the system of f(R) gravity, numerous
authors have examined various parts of Bianchi type models [12]-[14]. As of
late anisotropic cosmological arrangements in $f(R)$ gravity have gotten in
[15].

Bianchi type models are anisotropic and homogeneous. Such models are nine
altogether, yet their grouping permits them to be divided into two classes.
Bianchi ($I$, $II$, $VII$, $VIII$ and $IX$) models are in class $A$ and
Bianchi ($III$, $IV$, $V$, $VI$ and $VII$) are in class $B$. Spatially
homogeneous cosmological models assume a significant job in clarifying the
structure and space properties of all Einstein field conditions and
cosmological arrangements. Bianchi type-$I$ axially symmetrical cosmological
models with a magnetic field investigated [16] and string cosmology in Bianchi
$III$ and $VI_{0}$ has discussed in [17]-[18]. Bianchi type-III cosmological
model in $f(R,T)$ hypothesis of gravity have acquired in [19], Bianchi
types-$I$ and $V$ cosmological models in $f(R,T)$ gravity have acquired
careful arrangements of [20] and [21] have considered another class of
cosmological models in $f(R,T)$ gravity. Recently, Bianchi type $VI_{h}$
universe model in $f(R,T)$ gravity has been studied by[22]. There are not many
studies in modified theories in the literature on the Bianchi form $VI_{h}$
metric. Then in this analysis we investigated the distribution of MSQM in
$f(R,T)$ gravitation theory for Bianchi $VI_{h}$ universe [23]. Many
experiments have been carried out in this theory to study the dynamical
aspects of anisotropic cosmological models [24]-[28].

Bianchi type $VI_{h}$ cosmological model with perfect fluid as an origin of
gravitational field is given in the theory of self-creation for various cases
of matter discussed by [29]-[30]. This addresses the model’s physical and
mathematical properties. The objective of the paper is to study the Bianchi
type $VI_{h}$ model as suggested by Harko et al.[10] as part of an extended
theory of gravity. In the present work, our inspiration is to build up a
general formalism to research these anisotropic models with a simulated time-
differing deceleration parameter. In earlier works [31]-[33], the idea of this
scale factor was already conceived. Time varying $DP$ is required to explain a
progress of the universe from a decelerated stage to a accelerated stage at an
ongoing age. The other purpose of this paper is to study the swampland
criteria in Bianchi models. In this way, the Swampland criteria can be
utilized to oblige the quintessence $DE$ models that begin from a scalar field
theory. In literature there are lots of works that address the Swampland
criteria and quintessence models [34]-[37]. Here now we investigate whether
the dark energy complies with the swampland criteria. The string swampland
criteria for a powerful field theory to be reliable with the string theory.

Right now explore the consistency of the dark energy with swampland measure.
Also we investigate the variation of dark energy condition of state parameter.
Here we compute the variations of $\omega$ with $z$ for dark energy model. A
comparable test for swampland parameters for the scalar quintessence model has
been performed in a prior work [38]. They have been considered a core of this
research scope and investigated the swampland criteria for the quintessence
dark energy scope. In doing so they have taken the trial limits by composing
the varieties of dark energy condition of state in the standard CPL
parameterization shape and afterward make an interpretation of it to acquire
an upper bound of a recreated condition of $\omega(z)$. Here we specified two
parameters $\omega_{0}$ and $\omega_{a}$, dark energy parameters, with the
recent available SNe Ia , BAO and plank (2018) observation data. Chevallier-
Polarski-Linder (CPL) parameterization [39]-[42], Jassal-Bagla-Padmanabhan
(JBP)[43], Barboza-Alcaniz parameterization [44] are PADE-I and PADE-II [45].
These are some well known and most used dark energy parameterization in this
series. We used these five well-known parameterization of dark energy, namely
CPL, JBP, BA, PADE-I and PADE-II in our model and also find the swampland
correspondence with our related model.

The paper is structured as follows: For Sec. $2$, in the context of $f(R,T)$
gravity, we have developed the basic field equations and derived the related
geometric parameters. Section $3$ describes the dynamics of the model.
Stability and physical acceptability of the solution are discussed in Sec.
$4$. Parameterization of dark energy is addressed in Sec. $5$. Section $6$
shows the correspondence of swampland criteria of dark energy. In the last
section $7$, the description and conclusion are given.

##  2 Metric and Field Equations

Right now in this section we talk about the formalism created in an
insignificantly coupling theory of $f(R,T)$ to investigate certain models. We
consider a Bianchi $VI_{h}$ space time

$ds^{2}=dt^{2}-A^{2}dx^{2}-B^{2}{e^{2x}}dy^{2}-C^{2}e^{2hx}dz^{2},$ (1)

where $A(t)$, $B(t)$ and $C(t)$ are functions of the cosmic time $t$. The
space-time exponent $h$ is taken the value as $-1,0,1$. Here, we considered
the $h=-1$ because of the significance of the metric that envisages an
isolated universe with invalid total energy and momentum [46][47].

$T_{ij}=(p+\rho)u_{i}u_{j}-\rho_{B}x_{i}x_{j}-pg_{ij}.$ (2)

Here $u^{i}x_{i}=0$ and $x^{i}x_{i}=-u^{i}u_{i}=-1$. The four velocity vector
of the fluid in a co-moving model is $u^{i}=\delta_{0}^{i}$. $x^{i}$ is
anisotropic fluid direction and orthogonal to $u^{i}$. Such as the Perfect
fluid and anisotropic fluid ($\rho_{B}$), with energy density $(\rho)$. Harko
et al. [10] proposed the principle of $f(R,T)$ theory on the interaction of
matter and geometry.The four-dimensional Einstein-Hilbert movement is composed
as:

$S=\frac{1}{16\pi}\int d^{4}x\sqrt{-g}f(R,T)+\int d^{4}x\sqrt{-g})L_{m},$ (3)

where $f(R,T)$ is a function of $T(=g_{ij}T^{ij})$ and Ricci scalar $R$ in the
operation, $T^{ij}$ is the energy- momentum tensor. It is possible to take
Lagrangian $L_{m}$ as either $L_{m}=-p$ or as $L_{m}=\rho$. We consider a
minimal coupling of geometry and curvature for the modified gravity model,
assuming $f(R,T)=f(R)+f(T)$. We can write the field equations as [25]-[48]
after our earlier works.

$f_{R}(R)R_{ij}-\dfrac{1}{2}f(R)g_{i,j}+(g_{ij}\square-\nabla_{i}\nabla_{j})+f_{R}(R)=[8\pi+f_{T}(T)]T_{ij}+[pf_{T}(T)+\dfrac{1}{2}f(T)]g_{ij},$
(4)

where the corresponding partial differentiations are $f_{R}=\dfrac{\partial
f(R)}{\partial R}$ and $f_{T}=\dfrac{\partial f(T)}{\partial T}$. The field
equations [48] derive from a specific choice of $f(R,T)$=$\lambda(R+T)$

$G_{ij}=\left(\dfrac{8\pi+\lambda}{\lambda}\right)T_{ij}+\Delta(T)g_{ij},$ (5)

where $\lambda$ is a non-zero scaling variable in GR.
$G_{ij}=R_{ij}-\dfrac{1}{2}Rg_{ij}$ is the Einstein tensor. A time-dependent
effective cosmological constant can be defined with the variable
$\Lambda(T)=p+\frac{1}{2}T$ occurring in the field Eq. (5). Here $\Lambda(T)$
depends on the substance of the matter content and helps to accelerate.
Nevertheless, Eq. (5) has the same mathematical form of GR with a time varying
constant, due to the non-disappearing quantity of $\lambda$ it can not be
reduced to GR. Eq. (5), however, is a rescaled generalization of GR equations.
The field Eq. (5) can be written specifically for Bianchi form $VI_{h}$ space-
time in modified gravity as:

$\dot{H}_{y}+H_{y}^{2}+\dot{H}_{z}+H_{z}^{2}++H_{y}H_{z}+\frac{1}{A^{2}}=-\left(\frac{16\pi+3\lambda}{2\lambda}\right)\left(p-\rho_{B}\right)+\dfrac{\rho}{2},$
(6)

$\dot{H}_{x}+H_{x}^{2}+\dot{H}_{z}+H_{z}^{2}+H_{x}H_{z}-\frac{1}{A^{2}}=-\left(\frac{16\pi+3\lambda}{2\lambda}\right)p+\left(\dfrac{\rho_{B}+\rho}{2}\right),$
(7)

$\dot{H}_{x}+H_{x}^{2}+\dot{H}_{y}+H_{y}^{2}++H_{x}H_{y}-\frac{1}{A^{2}}=-\left(\frac{16\pi+3\lambda}{2\lambda}\right)p+\left(\dfrac{\rho_{B}+\rho}{2}\right),$
(8)

$H_{x}H_{y}+H_{y}H_{z}+H_{x}H_{z}-\frac{1}{A^{2}}=\left(\frac{16\pi+3\lambda}{2\lambda}\right)\rho-\left(\dfrac{p-\rho_{B}}{2}\right),$
(9) $H_{y}-H_{z}=0.$ (10)

Here $\lambda$ is the non zero scale factor. Directional Hubble parameters for
the anisotropic model are $H_{x}=\frac{\dot{A}}{A}$, $H_{y}=\frac{\dot{B}}{B}$
and $H_{z}=\frac{\dot{C}}{C}$. In view of Eq. (10) we have $H_{y}=H_{z}$.
Assuming $H_{x}=mH_{z}$ for $m\neq 1$, $H$ is the mean Hubble parameter can be
defined as:

$H=\frac{\dot{a}}{a}=\frac{1}{3}(H_{x}+H_{y}+H_{z})=\frac{1}{3}\left(\frac{\dot{A}}{A}+\frac{\dot{B}}{B}+\frac{\dot{C}}{C}\right).$
(11)

By using Eq. (10), we get

$H=\frac{\dot{a}}{a}=\frac{1}{3}(H_{x}+2H_{y})=\left(\frac{m+2}{3}\right)H_{z}.$
(12)

Here $q$ is as a linear function of $H$ then

$q=-\frac{a\ddot{a}}{\dot{a}^{2}}=\frac{\beta}{\sqrt{2\beta t+k}}-1.$ (13)

In addition, as indicated by SNe Ia’s despise resent observation, the present
universe is extending and the estimation of DP is in the scope of $-1<q<0$
sooner or later. This is the reason our model is perfect with resent
observation, in every one of the three circumstances.

$q_{0}=-1+\beta H_{0}=-1+\frac{\beta}{\sqrt{2\beta t_{0}+k}}.$ (14)

Since the present estimation of declaration parameter can be taken as here $k$
and $\beta$ are positive constants. From Eq. (13) we observed that $q>0$ for
$\frac{\beta}{\sqrt{2\beta t+k}}<1$ and $q<0$ for$\frac{\beta}{\sqrt{2\beta
t+k}}>1$. so we can choose a value of $\beta$ and $k$.

Integrating Eq. (13) then we get a scale factor (a) which is depends on time.

$a(t)=e^{\frac{1}{\beta}\sqrt{2\beta t+k}}.$ (15)

We can express the directional Hubble parameters as
$H_{x}=e^{\frac{3m}{\beta(m+2)}\sqrt{2\beta t+k}}$ and
$H_{y}=H_{z}=e^{\frac{3}{\beta(m+2)}\sqrt{2\beta t+k}}$,
$A=e^{\frac{3m}{\beta(m+2)}\sqrt{2\beta t+k}}$,
$B=C=e^{\frac{3}{\beta(m+2)}\sqrt{2\beta t+k}}$,

so that the metric (1) can be written as

$ds^{2}=dt^{2}-(e^{\frac{6m}{\beta(m+2)}\sqrt{2\beta
t+k}})dx^{2}-(e^{\frac{6}{\beta(m+2)}\sqrt{2\beta
t+k}})e^{2x}dy^{2}-(e^{\frac{6}{\beta(m+2)}\sqrt{2\beta t+k}})e^{2hx}dz^{2}.$
(16)

The spatial volume $V$ is given as

$V=(AB^{2})=a^{3}(t).$ (17)

The Hubble parameter is defined as:

$H=\frac{1}{3}(H_{x}+2H_{y})=\dfrac{1}{\sqrt{2\beta t+k}}.$ (18)

The scalar expansion in the universe is

$\theta=3H=(H_{x}+2H_{y})=\dfrac{3}{\sqrt{2\beta t+k}}.$ (19)

The shear scalar is

$\sigma^{2}=3\left(\dfrac{m-1}{m+2}\right)^{2}\dfrac{1}{\sqrt{2\beta t+k}}.$
(20)

The cosmic jerk parameter $j$ in cosmology is characterized as the third
derivative of the scale factor concerning the astronomical time is given as

$j=\frac{1}{H^{3}}\left(\frac{\dddot{a}}{a}\right)=\left(q+2q^{2}-2\frac{\dot{q}}{H}\right).$
(21)

In cosmology the jerk parameter is used to describe the models close to
$\Lambda CDM$. It is acknowledged that for models with negative estimation of
the deceleration parameter and positive estimation of the jerk parameter, the
transition of the universe from decelerated stage to accelerated stage
happens. The $\Lambda CDM$ jerk parameter has consistent with $j=1$. Right
now, we get

$j=1-\frac{3\beta}{\sqrt{2\beta t+k}}+\left(\frac{3\beta^{2}}{{2\beta
t+k}}\right).$ (22)

Here we take three observational data and find the value of ($\beta$, $k$) to
be the best fit with the latest observations and considered for drawn all
figures.

##  3  Dynamics of the Model

In this section we can obtain physical and kinematic parameters from the Eq.
(6)-(11).

$\rho=\frac{6}{1-4\alpha^{2}}\left[\frac{2}{m+2}\left(\dot{H}+H^{2}\right)+\frac{(5-2m)-6\alpha(2m+1)}{(m+2)^{2}}H^{2}\right]+\frac{2a^{-\frac{6m}{m+2}}}{1-2\alpha}$
(23)

$\rho_{B}=\frac{6}{1-2\alpha}\left[\frac{m-1}{m+2}\left(\dot{H}+3H^{2}\right)\right]-\frac{4a^{-\frac{6m}{m+2}}}{1-2\alpha}$
(24)

$p=\frac{6}{1-4\alpha^{2}}\left[\frac{(m-1)+2\alpha(m+1)}{m+2}\left(\dot{H}+H^{2}\right)+\frac{(2m^{2}-4m-7)+2\alpha(2m^{2}+1)}{(m+2)^{2}}H^{2}\right]-\frac{2a^{-\frac{6m}{m+2}}}{1-2\alpha}$
(25)

The effective cosmological constant $\Lambda$ and EoS parameter ( $\omega$)
are other complex features of the model. Such parameters are calculated using
the scale function.

$\omega=\left[\frac{(1+2\alpha)\\{3(m^{2}+3m+2)(\dot{H}+H^{2})+(6m^{2}-18m-6)H^{2}\\}}{6(m+2)(\dot{H}+H^{2})-3(2m-5)H^{2}+(m+2)^{2}a^{-\frac{6m}{m+2}}-2\alpha]\left(9(2m+1)H^{2}-(m+2)^{2}a^{-\frac{6m}{m+2}}\right)}\right]-1$
(26)

$\Lambda=\frac{6}{(1+2\alpha)(m+2)}\left[\dot{H}+3H^{2}\right].$ (27)

Here we use $\alpha=(16\pi+3\lambda)/2\lambda$. All the physical parameters
are expressed in terms of Hubble parameter $(H)$ in the above equations.

(a) (b)

Figure 1: (a) Plot of DP ($q$) versus $t$ (b) Plot of DP ($q$) versus redshift $z$ Table 1: According to three observed data’s, the corresponding values of of DP $(q_{0})$ and Hubble parameter ($H_{0}$), we find the values of $\beta$ and k . Cases | Data | $q_{0}$ | $H_{0}$ | $\beta$ | $k$ | Reference
---|---|---|---|---|---|---
Case-I | Supernova type Ia Union | -0.73 | 73.8 | 0.0036 | 0.000084 | [49]
Case-II | BAO and CMB | $-0.54$ | $73.8$ | 0.0062 | 0.000016 | [50]
Case-III | OHD+JLA | -0.52 | 69.2 | 0.000069364 | 0.000008394 |  [51]-[52]

Figure $1(a)$ shows the assortment of deceleration parameter $(q)$ with
astronomical time $(t)$ as indicated by Eq. (13). We just observe our design
model is an acceleration stage ($q<0$) for $k=0.000084$ and $\beta=0.0036$
(case I). In (case-II) $k=0.000016$ and $\beta=0.0062$ our model depicts the
phase transition from +ve to -ve deceleration parameter$(q)$. This shows that
our model is changing from $(q>0)$ deceleration to $(q<0)$ acceleration.
$t_{c}=\dfrac{\beta^{2}-c}{2\beta}$ is the critical time at which the phase
transaction took place. Similarly, [51, 52] used the value of the joint
dataset (OHD+JLA) in which $k=0.0069364$ and $\beta=0.000008394$ we find
decelerating-accelerating phase transition (case III) (Table-1).

The average scale factor $a(t)$ as far as redshift $z$ is given by
$a(t)=\frac{a_{0}}{1+z}$, From Eq. (15), we get $\frac{\sqrt{2\beta
t+k}}{\beta}=ln(a)$, then $ln(a)=ln(a_{0})-ln(1+z)$ . Substituting the above
in Eq. (13), we shows $q$-parametrization given by

$q(z)=-1+\frac{1}{ln(a_{0})-ln(1+z)},$ (28)

where $a_{0}$ is present value of scale factor, for three cases of $(\beta,k)$
$(\beta=0.0036,k=0.000084,\beta=0.0062,k=0.000016,\beta=0.00006,k=0.00000)$
for plot the graphs. In the figure 1(b) we have discovered that $q(z)$
expansion is a smooth progress from a decelerated stage to accelerated period
of development and $q\to-1$ as $z\to-1$. As of late [53]-[54] have discovered
the change redshift from decelerating to accelerated in modified gravity
cosmology. SNe type Ia dataset has given the progress from past deceleration
to ongoing increasing speed at $\Lambda CDM$. More recently, in 2004 (HZSNS)
team have identified $z_{t}=0.46\pm 0.13$ at $(1\sigma)$ c.l. [2] which is
again improved to $z_{t}=0.43\pm 0.07$ at $(1\sigma)$ c.l. [7]. SNLS [55], and
recently complied by [56], provide a progress redshift $z_{t}+0.6(1\sigma)$ in
improved concurrence with the flat $\Lambda CDM$.

In addition, the $q(z)$ reconstruction is performed by the Joined
$(SNIa+CC+H_{0})$, which have acquired the change redshift
$z_{t}=0.69^{+0.9}_{-0.6}$, $0.65^{+0.10}_{-0.07}$ and $0.61^{+0.12}_{-0.85}$
inside $(1\sigma)$ [57]. which are seen as well predictable with past outcomes
[58]-[62] including the $\Lambda CDM$ expectation $z_{t}\approx 0.7$. Another
constraint of change redshift is $0.60\leq z_{t}\leq 1.18$ ($2\sigma$ joint
examination) [63]. From the $H(z)$ data we find evidence as $z_{t}=0.720\pm
0.14$ for redshift. which is in acceptable concurrence with the [64] assurance
of $zt=0.72\pm 0.05$ as well as [65] assurance of $z_{t}=0.82\pm 0.08$ at
$1\sigma$ error. This is again improved as $z_{t}=0.72\pm 0.14$ at $68\%$ c.l.
which is in acceptable understanding. From the combination of $H(z)$ and SNIa
datasets, we note that the transition from deceleration to acceleration in the
BI expansion process takes place at a redshift of $z_{t}=0.57\pm 0.0037$ which
is in acceptable concurrence with the outcomes acquired [66, 67]. Therefore,
with reference to resent observation of SNe Ia, the present universe is
expanding and at some stage the value of DP is within the range of $-1<q<0$.
So, we observe that in all three cases, our model is aligned with resent
findings.

The transition redshift for our derived models for two cases (iii)
$\beta=0.000008394$, $k=0.0069364$ and (ii) $\beta=0.0062$, $k=0.000016$ are
found to be $z_{t}=1.954$ and $z_{t}=2.28$ respectively (Fig. $1(b)$) which
are in good agreement with observational values.

(a) (b)

Figure 2: Plot of energy density $\rho$ versus $t$ and redshift. Here
$\lambda=1$, $\alpha=26.6$, $m=0.004$.

Figure $2(a,b)$ which is related to the Eq. (28), depict the energy density
$\rho$ with time $t$ and redshift $z$ for three cases. It is seen that $\rho$
remains positive during the infinite development. Here we additionally noticed
that $t\to 0$, $\rho\to\infty$ showing the big-bang situation. Our model is
likewise steady with ongoing perceptions. We conclude that in fig.2(a) $\&$
fig.2(b) the $\rho$ is a positive diminishing function of time and increasing
function of $z$. It approaches to zero as $t\to\infty$. Which is consistent
with the well establishment scenario.

(a) (b)

Figure 3: Plot of anisotropic fluid energy density $(\rho_{B})$ versus $t$ and
redshift. Here $\lambda=1$, $\alpha=26.6$, $m=0.004$.

Figure $3$ displays anisotropic fluid energy density in terms of cosmic time
$t$ and redshift $z$. Eq. (29) corresponding to anisotropic fluid energy
density $(\rho_{B})$ is a declining time function and remains positive during
the cosmic assessment. First, we can see in the figure 3(a), it fell sharply,
then gradually, and at the present epoch, it approached to a small positive
value. Here $(\rho_{B})$ tends to 0 as $t\to\infty$. But in Fgure 3(b)
anisotropic fluid energy density increases with redshift because density
decreases imply the volume increase, the expansion of the universe. In all
three cases that the anisotropic density of the dark matter decreases with
time and tends to zero at late times.

(a) (b)

Figure 4: Plot of pressure $p$ versus $t$ and redshift. Here $\lambda=1$,
$\alpha=26.6$, $m=0.004$.

Figure $4$ represents the expansion for fluid pressure $p$ for the model and
corresponding to (25). We observed that for all three cases pressure is
negative increasing function of time $t$ and redshift $z$. From the figures
$4(a)$ and $4(b)$ we observed for the homogeneous and isotropic model,
pressure is consistently negative and approaches zero at late time.

(a) (b)

Figure 5:  Plot of EoS parameter $\omega$ versus $t$ and redshift. Here
$\lambda=1$, $\alpha=26.6$, $m=0.004$.

Figs. 5(a) and 5(b) depict the behavior of EoS parameter $\omega$ with respect
to time $t$ and redshift $z$ respectively. The EoS parameter may lies in
phantom region $(\omega<-1)$. Both figures represent the same scenario of the
universe for all cases. It is clearly shows in fig. 5(a) that first equation
of state parameter decreased sharply and approach to a small negative values
at the present epoch. The fig. 5(b) shows that the EoS parameter is a
increasing function of $z$, the rapidity of its growth at the early stage
deponds on the types of the universe.

(a) (b)

Figure 6: Plot of cosmological constant $\Lambda$ versus $t$ and redshift.
Here $\lambda=1$, $\alpha=26.6$, $m=0.004$.

Figure 6(a) and 6(b) depicts the cosmological constant with respect to time
and redshift $z$. In all cases $\Lambda$ is decreasing function of time and
increasing function of redshift $z$. We expect that in the universe, the
positive value of $\Lambda$ the expansion will tends to accelerate. In fig.
6(a) we observed that the $\Lambda$ decreases more sharply as time increases
in empty universe as compared to radiating dominated and stiff fluid universe.
Here fig.6(b) shows that $\Lambda$ increases with high redshift. A positive
cosmological constant resists attractive gravity of matter due to its negative
pressure.

##  4  Physical Acceptability and Stability of the Outcomes

Now, we shall test by means of some recently used diagnostic tools, whether
our models are stable or not ?

###  4.1 Energy Condition

In this subsection, for our derived model, we are checking the energy
conditions. The three energy conditions (NES)$(\rho+p\geq 0)$,
(SEC)$(\rho+3p)\geq 0$ and (DEC)$(\rho-p)\geq 0$ are given. In Figs. $7(a)$,
$7(b)$ and $7(c)$, we have plotted the energy conditions with respect to
cosmoc time for all three cases I, II & III. From these figures we observe
that energy conditions violet in all cases which is aspected in the case of
DE.

(a)  (b)  (c)

Figure 7: Plot of energy conditions (NEC,SEC,DEC) with $t$. Here $\lambda=1$,
$\alpha=26.6$, $m=0.004$.

###  4.2 Velocity of Sound

Numerous authors [68]-[70] have researched the requirements on sound speed of
dynamic DE models with time-changing EoS ($\omega$) and presumed that
imperative on the sound speed of DE is exceptionally weak. DE parameters just
as other cosmological parameters are independent of the compelling sound speed
of DE, there is no limitation from present astronomical information on the
effective sound speed. The system is unstable if squared speed of sound
$v_{s}^{2}<0$. It is necessary that the $v_{s}^{2}$ sound speed should be less
than the velocity of light $c$. As we work with unit speed of light in
gravitational units, i.e. sound velocity exists within the range $0\leq
v_{s}^{2}= (\frac{dp}{d\rho})\leq 1$. We get the velocity of sound as,

$v_{s}^{2}=\frac{6}{1-4\alpha^{2}}\left[{\frac{(m-1)+2\alpha(m+1)}{m+2}}\left(\frac{-2\beta}{(2\beta
t+k)^{2}}+\frac{3(\beta)^{2}}{(2\beta
t+k)^{\frac{5}{2}}}\right)-2\beta\frac{(2m^{2}-4m-7)+2\alpha(2m^{2}+1)}{(m+2)^{2}(2\beta
t+k)^{2}}\right]$

$+12m\frac{4e^{\frac{-6m\sqrt{2\beta t+k}}{\beta(m+2)}}}{(m+2)\sqrt{2\beta
t+k}(1-2\alpha)}\times$
$\frac{1}{\frac{6}{1-4\alpha^{2}}\left[\frac{2}{m+2}\left(\frac{-2\beta}{(2\beta
t+k)^{2}}+\frac{3(\beta)^{2}}{(2\beta
t+k)^{\frac{5}{2}}}\right)-2\beta\frac{(-2m+5)+6\alpha(2m+1)}{(m+2)^{2}(2\beta
t+k)^{2}}\right]}$ $-12m\frac{4e^{\frac{-6m\sqrt{2\beta
t+k}}{\beta(m+2)}}}{(m+2)\sqrt{2\beta t+k}(1-2\alpha)}$ (29)

Figure 8: Plot of velocity of sound versus $t$. Here $\lambda=1$,
$\alpha=26.6$, $m=0.004$.

Figure 8 dipicts that a positive value of squared speed of sound $v_{s}^{2}>0$
represents a stable model. Here we notice that in case I, our model is
unstable as $v_{s}^{2}<0$ through whole evolution of the Universe but in case
II and case III, our model is stable in early phase of the Universe whereas
unstable in present phase.

##  5 Dark Energy Parameterization

Given a large number of scalar field models with a range of potentialities,
testing all individual models is always difficult. Alternatively, we often use
a parameterization of the evolution of dark energy that broadly describes a
large number of DE models in the scalar region. In parameterizing
$(\omega=p/\rho)$ dark energy equation of state is the most common practice.
So, as to evaluate the dynamical parts of the model, we have plotted the EoS
parameter $\omega$ as an element of redshift and the literature[71]-[82] also
contains a large number of parameterizations for $\omega$. The conduct of the
EoS parameter of the model has been contrasted and that of some notable EoS
parameterization like CPL [39]-[42], JBP[43], BA [44], PADE-I and PADE-II [45]
for our measurements of the evidence together with the regular $\Lambda CDM$
and the constant dark energy equation of state model.

Now, in this segment we consider the five well known DE parameterization. The
first is the Chevallier-Polarski-Linder model (CPL), where $\omega_{0}$ is the
present EoS value and its overall time evolution $\omega_{a}$ is the CPL model
written as follows.

$\omega=\omega_{0}+\omega_{a}\frac{z}{(1+z)}.$ (30)

The CPL parameterization issue at high redshift $z$ has been discussed in [71,
73]. To point out this behavior, the authors [71, 73] proposed CPL
parameterization where parameters of $\omega_{0}$ and $\omega_{a}$ have the
same definitions as those defined for parameterization of the CPL. Another we
consider that Jassal-Bagla-Padmanabhan (JBP) parameterization of DE as

$\omega=\omega_{0}+\omega_{a}\frac{z}{(1+z)^{2}}$ (31)

proposed in [43], this model represents a dark energy component in both low
and high redshift regions with rapid variation at low $z$ where the
parameterization of the CPL can not be generalized to the whole universe.
Here, the parameters $\omega_{0}$ and $\omega_{a}$ have the same meanings as
those defined for the above two models. Barboza-Alcaniz parameterization
proposed in [44], that this model presents a step forward in redshift areas
where the CPL parameterization can not be reached out to the whole history of
the universe. It is useful the structure is given by:

$\omega=\omega_{0}+\omega_{a}z\frac{1+z}{(1+z^{2})}$ (32)

which is well-behaved at $z\to{-1}$. The other parameterization of DE include
PADE-I and PADE-II The EoS parameter can be written in terms of $z$ as:

$\omega=\frac{\omega_{0}+\omega_{a}z\frac{z}{(1+z)}}{1+\omega_{b}\frac{1+z}{(1+z)}}$
(33)

here the EoS parameter with $\omega_{b}\neq{0}$ maintains a strategic distance
from the dissimilarity at $a\to\infty$ (or proportionally at $z=-1$) [45].

$\omega_{z}=\begin{cases}{\frac{\omega_{0}+\omega_{a}}{1+\omega_{b}}}~{}~{}~{}~{}for~{}~{}~{}a\to
0~{}~{}~{}~{}(z\to\infty~{}~{}early~{}time)\\\
{\omega_{0}=0~{}~{}for~{}~{}~{}a=1~{}~{}(z=0~{}~{}present~{}~{}time)}\\\
{\frac{\omega_{0}}{\omega_{b}}~{}~{}for~{}~{}a\to\infty~{}~{}~{}(z\to-1~{}~{}future~{}~{}time)}\\\
\end{cases}$

Here we have to set $\omega_{b}\neq{0}$ and ${-1}$. In this manner we conclude
that the PADE (I) formula is a well-behaved function in the scope of
$\leq{a}\leq\infty$ (or comparably at $-1\leq{z}\leq\infty$). Clearly PADE (I)
estimation has three free parameters: $\omega_{0},~{}\omega_{a},$ and
$\omega_{b}$, to evade singularities in the cosmic extension, $\omega_{b}$
necessities to lie in the range $-1<\omega_{b}<0$.
Here the present parameterization is composed as a component of $lna$. Right
now, EoS parameter can be composed as

$\omega=\frac{\omega_{0}+\omega_{a}ln\frac{1}{(1+z)}}{1+\omega_{b}ln\frac{1}{(1+z)}},$
(34)

where $\omega_{0},\omega_{a},$ and $\omega_{b}$ are constants [45] In PADE
(II) parameterization, to keep away from singularities at these ages, we need
to force $\omega_{b}\neq 0$

$\omega_{z}=\begin{cases}{\frac{\omega_{a}}{\omega_{b}}}~{}~{}~{}~{}for~{}~{}a\to
0~{}~{}~{}(z\to\infty~{}~{}early~{}~{}time)\\\
{\omega_{0}=0~{}~{}~{}for~{}~{}a=1~{}~{}(z=0~{}~{}present~{}~{}time)}\\\
{\frac{\omega_{a}}{\omega_{b}}~{}~{}for~{}~{}a\to\infty~{}~{}(z\to-1~{}~{}future~{}~{}time)}\\\
\end{cases}$

Table 2: Dark energy parameterization with best fit values of $\omega_{0}$ and $\omega_{a}$. Model | Parameterization | best fit parameter using SNe Ia LA | Reference
---|---|---|---
CPL | $\omega=\omega_{0}+\omega_{a}\frac{z}{(1+z)}$ | $\omega_{0}=-0.991\pm 0.036~{}~{}\omega_{a}=0.297\pm 0.779$ |  [39],[55]
$JBP$ | $\omega=\omega_{0}+\omega_{a}\frac{z}{(1+z)^{2}}$ | $\omega_{0}=-1.013\pm 0.070~{}~{}\omega_{a}=-0.297\pm 4.306$ |  [43] [55]
$BA$ | $\omega=\omega_{0}+\omega_{a}z\frac{1+z}{(1+z^{2})}$ | $\omega_{0}=-0.997\pm 0.034~{}~{}\omega_{a}=-0.245\pm 0.545$ |  [44] [55]
$PADE-I$ | $\omega=\frac{\omega_{0}+\omega_{a}z\frac{z}{(1+z)}}{1+\omega_{b}\frac{1+z}{(1+z)}}$ | $\omega_{0}=-0.825\pm 0.091~{}~{}\omega_{a}=-0.683\pm 0.040$ |  [45] [55]
$PADE-II$ | $\omega=\frac{\omega_{0}+\omega_{a}ln\frac{1}{(1+z)}}{1+\omega_{b}ln\frac{1}{(1+z)}}$ | $\omega_{0}=-0.889\pm 0.080~{}~{}\omega_{a}=0.297\pm 0.779$ |  [45] [55]

Figure 9 shows the behaviour of the EoS parameter verses redshift $z$ for five
DE parameterization. All the parameterization, contain two parameters
$\omega_{0}$ and $\omega_{a}$ and the the parameterization (PADE-1,PADE-II)
containts three parameters. Here $\omega_{0}$ is related to the present value
of the equation of state for the dark energy and $\omega_{a}$ $\&$
$\omega_{b}$ determines its evolution with time. In this framework, using the
best-fit values, we found that only the PADE-II parameterization remains in
the quintessence regime $(1<\omega<1/3)$ and at high redshifts, while they
enter in the phantom regime. The rest of the parameterizations(PADE-1, CPL,
BA, JBP) evolve in the phantom region $(\omega<-1)$ at high redshifts.

Figure 9: Plot of EoS parameter $\omega$ versus $z$ for five DE
parameterization

##  6 Correspondence of Quintessence Dark Energy Model with Swampland Criteria

Swampland criteria may also be examined for alternative Gravity theories [83,
84]. It is particularly interesting because some modified theories are a
useful paradigm to cure shortcomings of General Relativity at ultraviolet and
infrared scales, due to the lack of a full quantum gravity theory [85]. In
this direction Bebetti and Capozzilla [85] worked on the Noether Symmetry
method with Swampland conjecture in f(R) gravity. The selected f(R) models,
satisfying the swampland conjecture, are consistent, in principle, with both
early and late-time cosmological behaviors. In this section we are discussing
the consequences of swampland’s DE principles in the framework of current and
prospective cosmological observations. The swampland refers to a set of low-
energy theories which look consistent from low- energy perspective but fail to
be UV completed with quantum gravity.

The swampland Conjecure concerning the effective potential $V(\phi)$ of the
scalar field ( $\phi$) are provided by subsequent inequalities:

$Conjecure~{}~{}{1}~{}~{}~{}~{}~{}:~{}~{}M_{p1}\mid\triangledown V\mid\geq
c_{1}V$ (35)

Here $M_{p1}$ is the reduced planck mass. The field should not, additionally,
fluctuate more than around one planck unit over the entire universe’s history.

$Conjecure~{}~{}{2}~{}~{}~{}~{}~{}:~{}~{}\triangle\phi\lesssim c_{2}M_{p1},$
(36)

where $c_{1}$ and $c_{2}$ are the constants of first order has become
significant,

or otherwise light fields become important and the effective field theory
becomes invalid. The swampland criteria have started expansions proposed in
[65]. Recently lots of activity, due to their suggestions for inflation are
discussed in [86]-[95].

The gravitational field and the scalar field are depicted by the action

$S=\int
d^{4}x\sqrt{-g}\left(\frac{Mp^{2}}{2}{(R)}+L_{m}-\frac{1}{2}g^{ij}\partial_{i}\phi\partial_{j}\phi-V(\phi)\right),$
(37)

where $R$ stands for Ricci scalar, and $V(\phi)$ is a potential and $\phi$ is
the scalar field.

In this section we focus on the exponential quintessence model with a
potential of the form $V(\phi)=V_{0}e^{\lambda_{1}\phi}$ [93] We compute the
variations of dark energy equation of state for
$\lambda_{1}=M_{p1}V^{{}^{\prime}}/V$ and compare them with the dark energy
parametrisation. The behavior for the the pressure and the energy density of
the quintessence field [96] is given as,

$\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi),~{}~{}p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)$
(38)

Therefore the string theory criteria can be used to constrain the dark energy
models that originate from a scalar field theory. Numerous works in literature
that address Swampland criteria and quintessence models [87]-[92]. The string
Swampland criteria for an effective field theory to be consistent with the
string theory. The quintessence models of dark energy were significantly
compelled by the string theory of swampland criteria.

The behaviour of $\phi$ and $V(\phi)$ are shown in Figs. $10$ and $11$. It is
obvious that the scalar field $\phi$ is the increasing function verses $z$ and
potential field $V(\phi)$ in the terms of the scalar field $\phi$ are also a
positive decreasing function associated with swampland criteria 1 $\&$ 2.

The swampland criteria gives tight limitations to the dark energy models of
the late time acceleration of the Universe just as on in stationary models of
the early universe [97]-[99] right now, the basis of present and future
cosmological perceptions.

Figure 10: Plot of scalar field $(\phi)$ versus $z$. Here $\lambda=1$,
$\alpha=26.6$, $m=0.004$.  Figure 11: Plot of $V(\phi)$ versus $\phi$. Here
$\lambda=1$, $\alpha=26.6$, $m=0.004$. Figure 12: Plot of EoS
$(\omega_{upperbound})$ versus $z$ with CPL parameterization .

A discussion on the significance of exponential type of potential to address
the validity of swampland models is given in [100]. Dark energy has likewise
been considered in [94] for dark energy equation of state investigation with
CPL parametrization with regards to examining swampland measures. In figure 12
we plot equation of state as a function of red-shift. We further analyze our
dark energy with the 1$\sigma$, 2$\sigma$, 3$\sigma$ upper limits of $\omega$,
developed by CPL parameterization, with the recent cosmological results given
in [100]. We are calculating our results of $\omega$ vs z for standard
quintessence dark energy model for values of $\lambda_{1}$ where
$\lambda_{1}=0.6,0.8,1.0,1.2,1.4$ . In this figure if $\lambda_{1}=0.8$ the
quintessence DE model lies below in the 2$\sigma$ and 3$\sigma$ upper bounds
and $\lambda_{1}=1,1.6$ quintessence DE model lies below in the 3$\sigma$
upper bounds and and satisfied the Swampland criteria. Therefore the
quintessence dark energy model is fulfills and satisfied the swampland
criteria for recent data. Hence we have examined the results of swampland
parameters on scalar DE field models, to be specific general core.

##  7 Conclusions

In this paper we have presented anisotropic Bianchi Type$-VI_{h}$ space-time
filled with anisotropic fluid in the frame work of $f(R,T)$ gravity propose by
[35]. The motive is to obtain a set of field equations with a time-dependent
$\Lambda$.

The main features of the models are as follows:

  * •

We have discussed our theoretical models based on three data sets: (i)
supernova type Ia union data [49], (ii) BAO and CMB data [50] and (iii)
current data in combination with HOD and LA observations [51, 52].

  * •

The solution of the corresponding field equations is obtained by assuming a
time-dependent DP $q(t)=-1+\frac{\beta}{\sqrt{2\beta t+k}}$, where $k$ and
$\beta$ are arbitrary integrating constants. In plotting all figures, we have
used three sets of ($\beta,k$): (i) $\beta=0.0036$, $k=0.000084$, (ii)
$\beta=0.0064$, $k=0.000016$ and (iii) $\beta=0.000008394$, $k=0.0069364$
respectively (Table-1). These three sets of values of ($\beta,k$) are obtained
by using three observational data sets [49]-[52].

  * •

The transition redshift for our derived models for two cases (iii)
$\beta=0.000008394$, $k=0.0069364$ and (ii) $\beta=0.0062$, $k=0.000016$ are
found to be $z_{t}=1.954$ and $z_{t}=2.28$ respectively (Fig. $1(b)$) which
are in good agreement with observational values [101]-[106]. The current
$H(z)$ data redress a big redshift range, $0.7\leq z\leq 2.36$ [101]-[106]
larger than that covered by Type Ia supernovae ($0.01\leq z\leq 1.30$). The
$H(z)$ data can be used to trace the cosmological deceleration-acceleration
transition [60, 107].

  * •

For two cases (ii) $\beta=0.0062$, $k=0.000016$, and (iii)
$\beta=0.000008394$, $k=0.0069364$, the DP varies from deceleration (early
time) to acceleration (present time) whereas for another case (i)
$\beta=0.0036$, $k=0.0000084$, we find only accelerating universe (Fig.
$1(a)$).

  * •

For all the three cases, the energy density ($\rho$) is found to be a
decreasing function of time and remains always positive in the whole evolution
of the Universe (Fig. $2(a)$). The Fig. $2(b)$ shows the variation of $\rho$
with redshift ($z$). From this Fig. $2(b)$ we observe that $\rho$ is an
increasing function of $z$ which is consistent with well established law.

  * •

The anisotropic fluid energy density ($\rho_{B}$) is declining time function
and remains always positive (Figs. $3(a)$). We also observe that $\rho_{B}$ is
an increasing function of redshift (Figs. $3(b)$) for all three cases.

  * •

The fluid pressure ($p$) is negative increasing function of cosmic time $t$
and remains always negative (Figs $4(a)$) which show the existence of dark
energy (negative pressure). The $p$ is decreasing function of redshift (Figs
$4(b)$).

  * •

We observe that for one case (ii) the universes are varying in quintessence
$\omega\geq{-1}$ region [108] through of evolution, while later on crosses PDL
($\omega=-1$) and finally approaches to phantom region $\omega\leq{-1}$ [109]
(Figs. $5(a)$ & $(b)$). In other two cases (i) & (iii) the universe is varying
only in phantom region. Therefore, we conclude that we are living in phantom
scenario.

  * •

The latest cosmological observations in our derived model confirm the
existence of a decaying vacuum energy density of $\Lambda(t)$. These results
on type Ia supernova’s magnitude and redshift suggested that our universe
could extend through the cosmological $\Lambda$-term with induced cosmological
density. From Figs. $6(a)$ & $(b)$, we observe that the $\Lambda(t)$ is a
positive decreasing function of cosmic time ($t$) in all three cases models of
the Universe which is consistent with observations.

  * •

We have found that all energy conditions i.e. WEC, SEC and DEC are not
satisfied for all the universes for all three values of ($k,\beta$) (see Figs
(7a, b & c)). This is consistent with dark energy scenario.

  * •

We have observed that speed of sound remains less than light velocity ($c=1$)
throughout the universe evolution $(Figs.8)$ in the three cases (i), (ii) &
(iii). This prove the physical acceptability of our solutions.

  * •

The parameterization of DE EoS ($\omega$) is significant to evaluate the
dynamical part of the models. In Figs. ($9$), some renowned EoS
parameterization like CPL, JBP, BA, PADE-I and PADE-II have been graphed. The
results of our analysis of DE parameterization with the best-fit values of
$\omega_{0}$ and $\omega_{a}$ have been given in Table-2. Based on this
analysis, we put constraints on the model parameters and found that the
expansion data. In the present framework, using best-fit values, we found that
only PADE-II and JBP parameterization remains in the quintessence regime and
the rest CPL, BA and PADE-I evolve in the phantom region.

  * •

We have discussed the outcomes of swampland’s DE precepts in the reference of
current cosmological observations. The behaviour of $V(\phi)$ and $\phi$ are
shown in Figs. $10$ & $11$ respectively. We observe that potential is positive
declines from finite value and disappears at late time (present era). The
scalar field $\phi$ increases with time and always positive. In figure 12 we
plot equation of state as a function of red-shift, in this figure we have
analyze the dark energy with the 1$\sigma$, 2$\sigma$, 3$\sigma$ upper limits
of $\omega$, developed by CPL parameterization, with the recent cosmological
results.

Thus, our newly constructed models and their solutions are physically
acceptable. Therefore, for the better understanding of the characteristics of
Bianchi type-$VI_{h}$ cosmological models in our universe’s evolution within
the framework of $f(R,T)$ gravity theory and confrontation observational data,
may be helpful.

## Acknowledgment

A. Pradhan thanks to the Inter-University Centre for Astronomy & Astrophysics
(IUCAA), India for its supportive assistant under visting associateship
program. The authors are grateful to Prof. Y. Akrami and Dr. Umesh K. Shrama
for their fruitful comments on Section 6 which improves the paper in present
form.

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# Chasing the Threshold Bias of the 3-AP Game

Albert Cao, Felix Christian Clemen, Sean English, Xiaojian Li,
Tatum Schmidt, Leeann Xoubi, Weian Yin

###### Abstract

In a Maker-Breaker game there are two players, Maker and Breaker, where Maker
wins if they create a specified structure while Breaker wins if they prevent
Maker from winning indefinitely. A $3$-term arithmetic progression, or $3$-AP,
is a sequence of three distinct integers $a,b,c$ such that $b-a=c-b$. The
$3$-AP game is a biased Maker-Breaker game played on $[n]$ where every round
Breaker selects $q$ unclaimed integers for every Maker’s one integer. Maker is
trying to select points such that they have a $3$-AP and Breaker is trying to
prevent this. The main question of interest is determining the threshold bias
$q^{*}(n)$, that is the minimum value of $q=q(n)$ for which Breaker has a
winning strategy. Kusch, Rué, Spiegel and Szabó initially asked this question
and proved $\sqrt{n/12-1/6}\leq q^{*}(n)\leq\sqrt{3n}$. We find new strategies
for both Maker and Breaker which improve the existing bounds to

$(1+o(1))\sqrt{\frac{n}{5.6}}\leq q^{*}(n)\leq\sqrt{2n}+O(1).$

##  1 Introduction

Positional game theory analyzes two-player games, such as the famous kid’s
game Tic-Tac-Toe or abstract games played on hypergraphs. Two-player games
often have a framework where one player tries to build a specific structure
while the other player tries to prevent this. Such games are called _Maker-
Breaker games_. For an excellent overview on Maker-Breaker games and more
generally positional games we refer the reader to the survey [8] by
Krivelevich and the books [3] by Beck and [5] by Hefetz, Krivelevich,
Stojakovic and Szabó. Formally, a $(p,q)$-Maker-Breaker game is defined as
follows.

Let $H=(V,E)$ be a hypergraph. The vertex set $V$ is called the _board_ and
the vertices will be referred to as _points_. There are two players, Maker and
Breaker, and on every turn, Maker first chooses $p$ points that previously
have not been chosen, and then Breaker chooses $q$ points that have not been
chosen before. The game ends once either Maker selects all points of an edge
$e\in E$, or the entire board has been selected without this happening. Maker
wins if they completely occupy all vertices of any edge $e\in E$, and Breaker
wins otherwise, i.e. if they manage to claim at least one element in every set
$e\in E$.

The study of Maker-Breaker games was popularized by Erdős and Selfridge [6]
who found a criterion which gives a winning strategy for Breaker. Since then,
a variety of Maker-Breaker games have received considerable attention in the
literature. One of the most famous games is the game played on the edge set of
$K_{n}$, where the winning sets are collections of edges corresponding to a
fixed subgraph $H$. In particular, the case for $H=K_{3}$, known as the
_triangle game_ , has been studied intensively [1, 4, 7]. In this paper, we
study the so-called $3$-AP game.

A _$k$ -term arithmetic progression_, or _$k$ -AP_, is a set of integers of
the form $a,a+d,...,a+(k-1)d$ for some $a\in\mathbb{Z}$ and integer $d>0$. The
$k$-AP game is a $(1,q)$-Maker-Breaker game played on the first $n$ integers
and the winning sets are the collection of $k$-APs, i.e. in terms of the
hypergraph definition of Maker-Breaker games, we are playing on the hypergraph
$H=(V,E)$ where

$V=[n]\quad\quad\text{and}\quad\quad E=\\{S\subseteq V\ |\ S\text{ forms a
}k\text{-AP}\\}.$

In each turn, Maker selects one unoccupied number and then Breaker selects $q$
unoccupied numbers from $[n]$. If Maker manages to select all the numbers in a
$k$-AP, then they win. Otherwise, Breaker wins. Define the _threshold bias_
$q^{*}=q^{*}(n)$ to be the minimum value of $q$ for which Breaker has a
winning strategy on $[n]$. Kusch, Rué, Spiegel and Szabó [9] asked to find the
threshold bias for the $3$-AP game and provided the following upper and lower
bounds:

$\sqrt{\frac{n}{12}-\frac{1}{6}}\leq q^{*}(n)\leq\sqrt{3n}.$

The lower bound is deduced from a general theorem by Beck (Theorem 2 in [2])
providing a criterion for Maker to win. Their Breaker strategy for the upper
bound is to block all $3$-AP’s containing Maker’s last choice and one of their
previous choices.

We improve both the upper and lower bounds by doing a more in-depth analysis
of the $3$-AP game and utilizing the particular structures of $3$-APs.

######  Theorem 1.1.

The threshold bias of the $3$-AP game on $[n]$ satisfies

$(1+o(1))\sqrt{\frac{n}{3+\frac{3}{2}\sqrt{3}}}\leq
q^{*}(n)\leq\sqrt{2n}+O(1).$

Note that $3+\frac{3}{2}\sqrt{3}\approx 5.6$. Theorem 1.1 reduces the
multiplicative gap from $6$ to roughly $3.35$. The organization of this paper
is as follows. In Section 2, we present the proof of the lower bound of
Theorem 1.1, and in Section 3, we present the proof of the upper bound of
Theorem 1.1.

##  2 Lower Bound

In this section, we provide a strategy for Maker, yielding the lower bound on
the threshold bias from Theorem 1.1. To motivate our strategy, we first give a
definition. During the game, any unoccupied integer that forms a 3-AP with the
points Maker has placed previously, we will call a _threat_. Note that if at
any time there are more threats on the board than Breaker has moves, Maker
will win. Let

$\displaystyle
q\leq(1+o(1))\sqrt{\frac{n}{3+\frac{3}{2}\sqrt{3}}}\quad\text{and}\quad
x=\frac{1}{2}+\frac{1}{2\sqrt{3}}.$

The constant $3+\frac{3}{2}\sqrt{3}$ will be the maximum value of an
elementary function $f(x)$ that will arise from our combinatorial argument.
The maximum of such function will be reached at the corresponding value
$x=\frac{1}{2}+\frac{1}{2\sqrt{3}}$.

Our strategy for Maker is as follows. In the first $\lceil xq\rceil$ rounds,
Maker will choose the smallest unoccupied integer of value larger than
$\left\lceil\frac{n}{3}\right\rceil$. We will show that after these $\lceil
xq\rceil$ rounds, Maker can choose an unoccupied point that creates over $q$
threats, and thus Maker wins the game.

Let $B$ and $M$ be the set of integers occupied by Breaker and Maker,
respectively, after round $\lceil xq\rceil$ was played. Let $\ell$ denote the
largest integer occupied by Maker. Note that

$\ell\leq\left\lceil\frac{n}{3}\right\rceil+1+(q+1)\left\lceil
xq\right\rceil\leq\frac{n}{3}+q^{2}\leq\frac{2n}{3}$

for sufficiently large $n$, because in each of the $\left\lceil
xq\right\rceil$ rounds $q+1$ integers are played. We split up the board
$[n]=J_{1}\cup J_{2}\cup J_{3}\cup J_{4}$ into $4$ intervals

$\displaystyle
J_{1}=\left[1,\left\lceil\frac{n}{3}\right\rceil\right]\cap\mathbb{N},\
J_{2}=\left(\left\lceil\frac{n}{3}\right\rceil,\ell\right]\cap\mathbb{N},\
J_{3}=\left(\ell,\left\lfloor\frac{2n}{3}\right\rfloor\right]\cap\mathbb{N}\
\text{and}\
J_{4}=\left(\left\lfloor\frac{2n}{3}\right\rfloor,n\right]\cap\mathbb{N}.$

We set $b_{i}:=|J_{i}\cap B|$, so that $b_{i}$ is the number of Breaker points
in the interval $J_{i}$. Note that $b_{2}+|M|=|J_{2}|=\ell-\lceil n/3\rceil$,
because all of Maker’s points are in $J_{2}$ and there is no unoccupied point
in $J_{2}$. After the first $\lceil xq\rceil$ rounds, Breaker has played
$q\lceil xq\rceil$ moves. Thus, $b_{1}+b_{2}+b_{3}+b_{4}=|B|=q\lceil
xq\rceil$.

For $i\in J_{3}\setminus B$, we define

$\displaystyle A_{i}:=\left\\{2y-i,2i-y\ \bigg{|}\ y\in M\right\\}$

to be the set of potential threats created by left and right reflections. Note
that $A_{i}\subseteq[n]$, because
$i,y\in\left(\left\lceil\frac{n}{3}\right\rceil,\left\lfloor\frac{2n}{3}\right\rfloor\right]$.
Every element in $A_{i}$ completes a $3$-AP with one of Makers initial $\lceil
xq\rceil$ moves and the point $i$. Note that $|A_{i}|=2\lceil xq\rceil$. If
$A_{i}\cap M\neq\emptyset$ for some value of $i$, then Maker wins the game by
playing the integer $i$. Therefore, we can assume that $A_{i}\cap M=\emptyset$
for all $i\in J_{3}\setminus B$.

Further, we can assume that $|A_{i}\setminus B|\leq q$ for each $i$, otherwise
Maker can win in two turns. Indeed, if $|A_{i}\setminus B|>q$, Maker can
occupy the integer $i$ and then Breaker will not be able to block all the
integers in $A_{i}\setminus B$. Maker then wins the game by playing an
unoccupied integer in $A_{i}\setminus B$. Therefore, we can conclude that
$|A_{i}\cap B|\geq 2\lceil xq\rceil-q\geq(2x-1)q$. Summing up $|A_{i}\cap B|$
for all $i\in I_{3}\setminus B$, we get

$\displaystyle\sum_{i\in J_{3}\setminus B}|A_{i}\cap
B|\geq\left|J_{3}\setminus B\right|(2x-1)q.$ (1)

We can use a double counting argument to upper bound the sum on the left side
of (1). For every $b\in B$ the sum adds up the number of times $b$ appears in
some $A_{i}$. This is at most $|M|=\lceil xq\rceil$ times for every $b\in B$.
Thus,

$\displaystyle\sum_{i\in J_{3}\setminus B}|A_{i}\cap B|$
$\displaystyle=\sum_{b\in B}\left|\left\\{i\in J_{3}\setminus B:b\in
A_{i}\right\\}\right|\leq|B|\lceil xq\rceil=q\lceil xq\rceil^{2}.$ (2)

Combining (1) with (2), we get

$\displaystyle\left(|J_{3}|-b_{3}\right)(2x-1)q\leq q(xq+1)^{2}.$ (3)

Using that $\ell=\lceil n/3\rceil+b_{2}+|M|=\lceil
n/3\rceil+b_{2}+O(\sqrt{n})$, Inequality (3) can be written as

$\displaystyle\frac{n}{3}-b_{2}-O(\sqrt{n})=\left\lfloor\frac{2n}{3}\right\rfloor-\ell=|J_{3}|\leq
b_{3}+\frac{x^{2}}{2x-1}q^{2}+O(\sqrt{n}).$

Therefore,

$\displaystyle n\leq 3b_{2}+3b_{3}+\frac{3x^{2}}{2x-1}q^{2}+O(\sqrt{n})$
$\displaystyle\leq\left(3x+\frac{3x^{2}}{2x-1}\right)q^{2}+O(\sqrt{n})$
$\displaystyle\leq\left(3+\frac{3}{2}\sqrt{3}\right)q^{2}+O(\sqrt{n}),$

where we have used that $b_{2}+b_{3}\leq|B|\leq q\lceil xq\rceil$ and
$x=\frac{1}{2}+\frac{1}{2\sqrt{3}}$, which we chose to minimize the expression
on the right hand side. This contradicts
$q\leq(1+o(1))\sqrt{\frac{n}{3+\frac{3}{2}\sqrt{3}}}$. Thus, there exists an
$i\in J_{3}\setminus B$ such that $|A_{i}\setminus B|>q$ and therefore Maker
can win in two more rounds, completing the proof of the lower bound on the
threshold bias from Theorem 1.1. $\Box$

##  3 Upper bound

In this section, we provide a strategy for Breaker, yielding the upper bound
on the threshold bias from Theorem 1.1. Our Breaker strategy builds upon the
breaker strategy from Kusch, Rué, Spiegel and Szabó [9]. First, we will
briefly explain their strategy to help give some intuition for how our
strategy works.

Their Breaker strategy is simply to block all threats created on every turn.
Maker selects at most $\lceil n/(q+1)\rceil$ integers during the game and for
each pair of integers there are at most three $3$-AP’s containing them, i.e.
three threats. Thus, Breaker has to block at most

$3\left(\left\lceil\frac{n}{q+1}\right\rceil-1\right)\leq\frac{3n}{q}$

integers in each round to prevent Maker from winning. Thus, Breaker can block
Maker from winning (and consequently wins themselves) as long as
$\frac{3n}{q}\leq q$, or equivalently as long as $q\geq\sqrt{3n}$. This proves
$q^{*}(n)\leq\sqrt{3n}$.

Now, we will explain our strategy and show that it improves the upper bound
for the threshold bias. Let

$\displaystyle q\in\left[\sqrt{2n}+O(1),\sqrt{3n}\right].$ (4)

We classify Breaker’s moves into two different types: forced moves and free
moves. _Forced moves_ are moves that Breaker must play to block threats. _Free
moves_ are any moves that Breaker has after Breaker blocks all threats. Our
improvement over the pre-existing Breaker strategy is that we strategically
use our free moves to reduce the maximum number of threats that occur on any
round.

We partition the board $[n]=I_{1}\cup I_{2}\cup I_{3}$ into three consecutive
intervals

$\displaystyle
I_{1}=\left[1,\left\lfloor\frac{n}{3}\right\rfloor\right]\cap\mathbb{N},\quad
I_{2}=\left(\left\lfloor\frac{n}{3}\right\rfloor,\left\lceil\frac{2n}{3}\right\rceil\right]\cap\mathbb{N},\quad\text{and}\quad
I_{3}=\left(\left\lceil\frac{2n}{3}\right\rceil,n\right]\cap\mathbb{N},\quad$

which we will refer to as _left_ , _middle_ and _right intervals_ ,
respectively. In our strategy, Breaker plays all their free moves arbitrarily
in the middle interval until the interval is entirely occupied. Let $t^{*}$ be
the first round in which all integers of the middle interval have been chosen
(note that it is not obvious that $t^{*}$ is well-defined as theoretically
Maker could win prior to the middle becoming full; we will show that this does
not happen). On turn $t^{*}$, after the middle interval is completely
occupied, Breaker might have some free moves left and can play those remaining
ones arbitrarily.

In turns $t>t^{*}$, Breaker will play all its free moves arbitrarily in the
interval that Maker played in on that turn. Under the assumption that Maker
does not win, eventually this will lead to one of $I_{1}$ or $I_{3}$ becoming
fully occupied, and on this turn, Breaker can play any remaining free moves
arbitrarily.

A key observation for our strategy is that after time $t^{*}$ any new Maker
point can create at most two (instead of three) threats with any previously
played Maker point in the middle, see Figure 1 for an illustration.

######  Claim 3.1.

Let $t>t^{*}$. Any new Maker point can create at most two threats with any
previously played Maker point in the middle interval, $I_{2}$

###### Proof.

Let $m\in[n]$ be a new Maker point played after time $t^{*}$ and
$m^{\prime}\in I_{2}$ be an integer occupied by Maker. First assume that $m\in
I_{1}$. Only the three integers $(m+m^{\prime})/2,2m^{\prime}-m$ and
$2m-m^{\prime}$ can be threats. Assume that $(m+m^{\prime})/2$ and
$2m^{\prime}-m$ are threats. We will show that $2m-m^{\prime}$ is not a
threat. Since $(m+m^{\prime})/2<m^{\prime}$, we have $(m+m^{\prime})/2\in
I_{1}$ to be a threat. Since $2m^{\prime}-m>m^{\prime}$, we have
$2m^{\prime}-m\in I_{3}$ to be a threat. Therefore,

$\displaystyle\frac{m+m^{\prime}}{2}\leq\left\lfloor\frac{n}{3}\right\rfloor\quad\text{and}\quad\left\lceil\frac{2n}{3}\right\rceil<2m^{\prime}-m\leq
n.$

Hence,

$\displaystyle
2m-m^{\prime}=(m+m^{\prime})+(m-2m^{\prime})<2\left\lfloor\frac{n}{3}\right\rfloor-\left\lceil\frac{2n}{3}\right\rceil\leq
0,$

and therefore $2m-m^{\prime}$ cannot be a threat. We conclude that
$m^{\prime}$ together with $m$ creates at most two threats. If $m\in I_{3}$,
the analysis is similar. ∎

$\frac{n}{3}$$\frac{2n}{3}$$1$$n$$\boldsymbol{\times}$$\boldsymbol{\times}$$\circ$$\circ$
Figure 1: Pairs of Maker points with one being in the middle interval create
at most two threats. $\times$’s represent Maker points, and $\circ$’s
represents the threats created by those points.

Another important observation is that any new Maker point cannot create any
threats with Maker points on the other side of the middle.

######  Claim 3.2.

Let $t>t^{*}$. A new Maker point cannot create a threat with a Maker point on
the other side of the middle.

###### Proof.

This claim simply follows from the fact that for $m\in I_{1}$ and
$m^{\prime}\in I_{3}$, we have
$(m+m^{\prime})/2,2m^{\prime}-m,2m-m^{\prime}\notin I_{1}\cup I_{3}$, because

$\displaystyle\frac{m+m^{\prime}}{2}\leq\frac{\left\lfloor\frac{n}{3}\right\rfloor+n}{2}\leq\left\lceil\frac{2n}{3}\right\rceil,\quad\frac{m+m^{\prime}}{2}>\frac{1+\left\lceil\frac{2n}{3}\right\rceil}{2}\geq\left\lfloor\frac{n}{3}\right\rfloor,$
$\displaystyle
2m^{\prime}-m>2\left\lceil\frac{2n}{3}\right\rceil-\left\lfloor\frac{n}{3}\right\rfloor\geq
n\quad\text{and}\quad
2m-m^{\prime}<2\left\lfloor\frac{n}{3}\right\rfloor-2\left\lceil\frac{2n}{3}\right\rceil\leq
0.$

∎

Now, let us get an upper bound on when turn $t^{*}$ occurs.

######  Claim 3.3.

We have $t^{*}\leq\frac{q}{3}$.

###### Proof.

On any turn $t$ there is a maximum of $3(t-1)$ forced moves so Breaker has a
minimum of $q-3(t-1)$ free moves on this turn. Thus, for any turn $t$ prior to
the middle interval being filled, Breaker will have played a total of at least

$\displaystyle\sum^{t}_{j=1}q-3(j-1)=tq-\frac{3}{2}(t^{2}-t)\geq
tq-\frac{3}{2}t^{2}$ (5)

free moves, all of which Breaker played in the middle interval. Since
$|I_{2}|\leq\frac{n}{3}+2$, once the total free moves Breaker has been able to
play exceeds $\frac{n}{3}+2$, the middle third is guaranteed to be completely
occupied. Thus, using (5), we conclude that

$t^{*}\leq\frac{q}{3}-\frac{1}{3}\sqrt{q^{2}-2n-12}\leq\frac{q}{3}.\qed$

Note that since $q-3t^{*}\geq 0$, Breaker will always have free moves before
time $t^{*}$ and in particular, Breaker does not lose the game before time
$t^{*}$.

Given any time $t$, let $m_{1}(t)$, $m_{2}(t)$ and $m_{3}(t)$ denote the
number of Maker points in the left, middle and right interval respectively,
and let $b_{1}(t)$, $b_{2}(t)$ and $b_{3}(t)$ denote the number of Breaker
points in the left, middle and right interval, respectively, after turn $t$
was completed.

Assume it is Breaker’s turn in some round $t>t^{*}$. We will show that Breaker
can occupy all threats in this round, which further implies that Breaker
eventually wins the game. Without loss of generality, Maker has played their
last move in say $I_{1}$ (if they play in $I_{3}$, the analysis is similar).
We claim that Maker has created at most

$3m_{1}(t)+2m_{2}(t)=3m_{1}(t)+2m_{2}(t^{*})$

threats. Indeed, there are at most $3m_{1}(t)$ threats that were made by the
new Maker point with old Maker points in the left interval, and then there are
at most $2m_{2}(t)=2m_{2}(t^{*})$ threats made by the new point with old Maker
points from the middle interval. There are no threats made with old Maker
points from the right interval, as all integers completing a $3$-AP are either
outside of the board or in the middle interval and therefore blocked already.
Thus, as long as

$3m_{1}(t)+2m_{2}(t^{*})\leq q,$ (6)

Breaker will block all threats created by Maker in round $t$.

In the remaining part of the proof, we show that (6) holds. To do so we
collect a few equalities and inequalities which will help us to upper bound
$m_{1}(t)$, the first term in (6). Note that for all $t$,

$\displaystyle m_{1}(t)+m_{2}(t)+m_{3}(t)=t\quad\text{ and }\quad
b_{1}(t)+b_{2}(t)+b_{3}(t)=qt$

because Maker occupies one integer and Breaker $q$ integers in each turn.
Breaker plays at most

$\displaystyle b_{3}(t^{*})\leq
3\binom{m_{3}(t^{*})}{2}+2m_{2}(t^{*})m_{3}(t^{*})+\binom{m_{2}(t^{*})}{2}+m_{1}(t^{*})m_{2}(t^{*})+O(\sqrt{n})$
(7)

points in the last third until round $t^{*}$, because a pair of Maker points
$b,b^{\prime}$ creates at most

  * •

3 threats in $I_{3}$ if $b,b^{\prime}\in I_{3}$,

  * •

2 threats in $I_{3}$ if $b\in I_{3},b^{\prime}\in I_{2}$,

  * •

1 threat in $I_{3}$ if $b,b^{\prime}\in I_{2}$,

  * •

1 threat in $I_{3}$ if $b\in I_{2},b^{\prime}\in I_{1}$,

  * •

0 threats in $I_{3}$ if $b,b^{\prime}\in I_{1}$,

  * •

0 threats in $I_{3}$ if $b\in I_{1},b^{\prime}\in I_{3}$.

In round $t^{*}$, Breaker played at most $q=O(\sqrt{n})$ integers in $I_{3}$,
explaining the $O(\sqrt{n})$-term in (7). In each of the turns between time
$t^{*}$ and $t$ where Maker plays in the first third (there are
$m_{1}(t)-m_{1}(t^{*})$ many of those turns) Breaker plays at most
$m_{2}(t^{*})$ moves in $I_{3}$. Those are the forced moves which come from
threats created by a new Maker point in the first third and Maker points in
the middle. In each of the turns between time $t^{*}$ and $t$ where Maker
plays in the last third, Breaker trivially plays at most $q$ moves in $I_{3}$.
Thus, between time $t^{*}$ and $t$, Breaker plays at most

$\displaystyle b_{3}(t)-b_{3}(t^{*})\leq
q\left(m_{3}(t)-m_{3}(t^{*})\right)+m_{2}(t^{*})\left(m_{1}(t)-m_{1}(t^{*})\right)+O(\sqrt{n})$
(8)

integers in the last third. Note that

$\displaystyle\left\lceil\frac{2n}{3}\right\rceil=|I_{1}\cup I_{2}|\geq
b_{1}(t)+b_{2}(t)=qt-b_{3}(t)=q(m_{1}(t)+m_{2}(t)+m_{3}(t))-b_{3}(t).$ (9)

Combining (7), (8) and (9), we get that

$\displaystyle q($ $\displaystyle
m_{1}(t)+m_{2}(t)+m_{3}(t))-3\binom{m_{3}(t^{*})}{2}-2m_{2}(t^{*})m_{3}(t^{*})-\binom{m_{2}(t^{*})}{2}-m_{1}(t^{*})m_{2}(t^{*})$
$\displaystyle-$ $\displaystyle
q(m_{3}(t)-m_{3}(t^{*}))-m_{2}(t^{*})(m_{1}(t)-m_{1}(t^{*}))\leq\frac{2n}{3}+O(\sqrt{n}).$
(10)

Inequality (3) can be simplified to

$\displaystyle m_{1}$
$\displaystyle(t)(q-m_{2}(t^{*}))+q(m_{2}(t^{*})+m_{3}(t))-\frac{3}{2}m_{3}(t^{*})^{2}-2m_{2}(t^{*})m_{3}(t^{*})-\frac{1}{2}m_{2}(t^{*})^{2}$
$\displaystyle-$ $\displaystyle
q(m_{3}(t)-m_{3}(t^{*}))\leq\frac{2n}{3}+O(\sqrt{n}).$ (11)

Rewriting (3), $m_{1}(t)$ can be upper bounded by

$\displaystyle\ \ \
m_{1}(t)\leq\frac{1}{(q-m_{2}(t^{*}))}\bigg{(}\frac{2n}{3}-q(m_{2}(t^{*})+m_{3}(t))+\frac{3}{2}m_{3}(t^{*})^{2}+2m_{2}(t^{*})m_{3}(t^{*})+\frac{1}{2}m_{2}(t^{*})^{2}$
$\displaystyle+q(m_{3}(t)-m_{3}(t^{*}))\bigg{)}+O(1)$
$\displaystyle=\frac{1}{(q-m_{2}(t^{*}))}\bigg{(}\frac{2n}{3}-qm_{2}(t^{*})+\frac{1}{2}m_{2}(t^{*})^{2}+m_{3}(t^{*})\left(\frac{3}{2}m_{3}(t^{*})+2m_{2}(t^{*})-q\right)\bigg{)}+O(1)$
$\displaystyle\leq\frac{1}{(q-m_{2}(t^{*}))}\bigg{(}\frac{2n}{3}-qm_{2}(t^{*})+\frac{1}{2}m_{2}(t^{*})^{2}\bigg{)}+O(1),$
(12)

where we used Claim 3.3 to get $\frac{3}{2}m_{3}(t^{*})+2m_{2}(t^{*})\leq
2t^{*}\leq\frac{2q}{3}$ in the last inequality. Finally, we upper bound the
left hand side of (6) using our bound on $m_{1}(t)$ from (3),

$\displaystyle 3m_{1}(t)+2m_{2}(t^{*})$
$\displaystyle\leq\frac{3}{(q-m_{2}(t^{*}))}\bigg{(}\frac{2n}{3}-qm_{2}(t^{*})+\frac{1}{2}m_{2}(t^{*})^{2}\bigg{)}+2m_{2}(t^{*})+O(1)$
$\displaystyle\leq\frac{3}{(\sqrt{2n}-m_{2}(t^{*}))}\bigg{(}\frac{2n}{3}-\sqrt{2n}m_{2}(t^{*})+\frac{1}{2}m_{2}(t^{*})^{2}\bigg{)}+2m_{2}(t^{*})+O(1).$
(13)

Let $c\geq 0$ be such that $m_{2}(t^{*})=c\sqrt{n}$. Since $m_{2}(t^{*})\leq
t^{*}\leq q/3$, we have $c<1$. Define

$f(x):=\frac{3}{\sqrt{2}-x}\left(\frac{2}{3}-\sqrt{2}x+\frac{x^{2}}{2}\right)+2x,$

and note that the right-hand side of (3) is $f(c)\sqrt{n}+O(1)$. The first
derivative of $f$ is non-positive for $0\leq x\leq 1$,

$\displaystyle\frac{df}{dx}=\frac{c(c-2\sqrt{2})}{2(\sqrt{2}-c)^{2}}\leq 0.$

Therefore, $f(c)\leq f(0)=\sqrt{2}$. We conclude

$\displaystyle 3m_{1}(t)+2m_{2}(t^{*})\leq
f(c)\sqrt{n}+O(1)\leq\sqrt{2n}+O(1)\leq q.$

Hence, inequality (6) holds, completing the proof of the upper bound on the
threshold bias from Theorem 1.1. $\Box$

##  4 Concluding remarks

We would like to remark that both strategies presented in this paper focus on
first occupying integers in the middle. This matches the intuition that those
integers are the most valuable as they are contained in the most $3$-AP’s. We
believe that optimal strategies will be in this spirit too.

A natural variant of the 3-AP Game on $[n]$ is the 3-AP Game on
$\mathbb{Z}/n\mathbb{Z}$, i.e. the winning sets are triples forming a 3-AP in
the cyclic group $\mathbb{Z}/n\mathbb{Z}$. Denote the threshold bias of this
game $q^{*}_{c}(n)$. We have

$(1+o(1))\sqrt{\frac{n}{3+\frac{3}{2}\sqrt{3}}}\leq q^{*}(n)\leq
q^{*}_{c}(n)\leq\sqrt{3n}.$ (14)

Note that $q^{*}(n)\leq q^{*}_{c}(n)$ holds trivially, since the winning sets
of the 3-AP game on $[n]$ can be viewed as subsets of the winning sets of the
3-AP game on $\mathbb{Z}/n\mathbb{Z}$ and therefore Maker will win on
$\mathbb{Z}/n\mathbb{Z}$ if they play with their winning strategy on $[n]$.

The upper bound in (14) follows from the same proof used in [9] (see the first
few paragraphs of Section 3 for a summary of this strategy), as the strategy
only relied on the fact that every pair of integers is in at most three
$3$-AP’s, which is true both in $[n]$ and $\mathbb{Z}/n\mathbb{Z}$. The upper
bound proof from Section 3 does not translate to this setting, since our
improvement over the bound in [9] comes specifically from exploiting the fact
that integers in $[n]$ near $1$ or $n$ are in less $3$-AP’s than the integers
in the middle of $[n]$ (which is not true for $\mathbb{Z}/n\mathbb{Z}$).

Another direction of further study could be to take a look at the more general
$k$-AP game. The right order of magnitude for the threshold bias was
established to be $n^{1/(k-1)}$ for any $k$ in [9], however they do not give
explicit constants. Note that

$\displaystyle q_{k}^{*}(n)\geq\left(\frac{n}{k(k-1)^{2}}\right)^{1/(k-1)}$

follows from a theorem by Beck (Theorem 2 in [2]).

Another interesting variation of the 3-AP Game is the _Schur game_. This is
the Maker-Breaker game on $[n]$ with the winning sets being _Schur triples_ ,
i.e. three distinct integers $a,b,c$ such that $a+b=c$. Denote $q^{*}_{s}(n)$
the threshold bias of the Schur game. Since any pair of integers can at most
be in two Schur triples, the bounds

$\displaystyle(1+o(1))\sqrt{\frac{n}{8}}\leq q^{*}_{s}(n)\leq\sqrt{2n}$ (15)

can be established quickly. The lower bound in (15) follows from Beck’s
theorem (Theorem 2 in [2]) and the upper bound follows from the same proof
used in [9] for the 3-AP game (see the first few paragraphs of Section 3 for a
summary of this strategy). The proof techniques we used for Theorem 1.1 can be
applied to get improvements on the bounds (15) for the Schur game.

##  5 Acknowledgments

The authors would like to thank the Illinois Geometry Lab for facilitating
this research project, and also would like to thank the anonymous referees,
whose comments helped improve the paper. The second author would like to thank
József Balogh for introducing this problem to him. This material is based upon
work supported by the National Science Foundation under Grant No. DMS-1449269.
Any opinions, findings, and conclusions or recommendations expressed in this
material are those of the authors and do not necessarily reflect the views of
the National Science Foundation.

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