# Probing the multiwavelength emission scenario of GRB 190114C

Marc Klinger1 , Donggeun Tak1,2 , Andrew M. Taylor1 , and Sylvia J. Zhu1
1Deutsches Elektronen-Synchrotron DESY, Platanenallee 6, 15738 Zeuthen,
Germany
2SNU Astronomy Research Center, Seoul National University, 1 Gwanak-rho,
Gwanak-gu, Seoul, Korea E-mail<EMAIL_ADDRESS>
(Accepted 2023 January 6. Received 2023 January 6; in original form 2022 July
15)

###### Abstract

Multiwavelength observation of the gamma-ray burst, GRB 190114C, opens a new
window for studying the emission mechanism of GRB afterglows. Its Very-High-
Energy (VHE; $\gtrsim 100$ GeV) detection has motivated an inverse Compton
interpretation for the emission, but this has not been tested. Here, we
revisit the early afterglow emission from 68 to 180 seconds and perform the
modeling likelihood analysis with the keV to TeV datasets. We compute for the
first time the statistical preference in the combined synchrotron (syn) and
synchrotron self-Compton (SSC) model over the syn-only model. In agreement
with earlier analyses, between 68 and 110 seconds an unstable preference for
the SSC model can be found, which can also be explained by systematic cross
calibration effect between the included instruments. We conclude that there is
no stable statistical preference for one of the two models.

###### keywords:

gamma-ray bursts, radiation mechanisms: non-thermal, acceleration of
particles, methods: data analysis

††pubyear: 2023††pagerange: Probing the multiwavelength emission scenario of
GRB 190114C–C

##  1 Introduction

Catastrophic events such as stellar core collapses can lead to the launching
of an ultrarelativistic jet. The decelerating shock front set up between this
jetted outflow and the surrounding circumburst medium redistributes part of
the jet’s kinetic energy into thermal heating, turbulent magnetic fields, and
non-thermal charged particles. The emission produced by the cooling of these
non-thermal particles can be observed up to gamma-ray energies and is known as
the gamma-ray burst (GRB) afterglow emission. The earlier time GRB emission,
referred to as the prompt emission, shows a complex phenomenology of temporal
features. In contrast, the afterglow evolves smoothly, both in time and in its
spectral energy distribution. The simplicity of this observable phenomena is
consistent with a simple underlying process driving the afterglow emission,
providing a clean test-bed to probe the particle acceleration process.

In the standard afterglow scenario, the non-thermal radio to GeV gamma-ray
emission is produced by leptonic synchrotron emission, whilst the highest
energy gamma-rays are expected to be produced by the inverse Compton
upscattering of the synchrotron photons by the parent electron population
(synchrotron self-Compton, SSC) (Kumar & Zhang, 2015). The recent detections
of Very-High-Energy (VHE, $>\\!100$ GeV) emission from GRBs (H.E.S.S.
Collaboration et al., 2019; MAGIC Collaboration et al., 2019a; H.E.S.S.
Collaboration et al., 2021) have offered an opportunity to test this scenario
for the very first time.

For GRB 190114C, the VHE emission was interpreted as requiring an additional
SSC component ( MAGIC Collaboration et al. (2019b), but also e.g. Wang et al.
(2019); Derishev & Piran (2021); Gill & Granot (2022)), although without a
statistical significance for this preference being provided. In contrast, the
hardness of the VHE emission of GRB 190829A significantly preferred an
extension of the synchrotron X-ray emission up to the VHE scale (H.E.S.S.
Collaboration et al., 2021). This apparent difference in the interpretations
for these two GRBs encourages a re-test of the preferred emission scenario for
GRB 190114C.

In this paper, we give a statistically robust basis to the question as to
whether a preference for an two-component SSC model over a single-component
extended synchrotron emission for GRB 190114C is found. We present a full
likelihood analysis of the early afterglow of GRB 190114C with the
multiwavelength dataset from X-rays to VHE gamma rays. We discuss the
afterglow radiation model used here in Section 2.1, followed by the data
reduction in Section 2.2 and the subsequent fitting in Section 2.3. Finally,
we present our results in Section 3, and discuss the conclusions that can be
drawn in Section 4.

##  2 Methods

###  2.1 Reduced afterglow radiation model

We consider a reduced SSC model, within which both the SSC model as well as an
extended synchrotron model can be treated as subclasses. Our reduced SSC
model’s photon spectrum consists of a synchrotron and an inverse Compton
component. We infer the two components as convolutions of the radiation
kernels for synchrotron111eq. 7.43 of Dermer & Menon (2009) and inverse
Compton scattering 222eq. 2.48 of Blumenthal & Gould (1970) implemented as in
the AM3 code (Gao et al., 2017), with a non-thermal electron spectrum in a
single acceleration and radiation zone. We use two phenomenological parameters
to separate between the two models: Firstly, the parameter $N_{\text{IC}}$
controls the ratio of the peak inverse Compton to synchrotron energy flux.
Secondly, the parameter $\eta$ controls the electron acceleration rate — in
units of the Bohm acceleration rate — which, in turn, dictates the maximum
synchrotron photon energy. Utilizing this parameterization, the parameter
range $N_{\text{IC}}>0$ and $\eta>1$ describes the SSC model, and the
parameter range $N_{\text{IC}}=0$ and $\eta<1$ describes the extended
synchrotron model.

The single radiation zone in our reduced SSC model is assumed to be the blast
wave behind the forward shock, as commonly defined in the fireball model (see
e.g. Kumar & Zhang (2015) for a review and references therein). Several
earlier works (e.g. Meszaros & Rees, 1993; Dermer et al., 2000; Zhang &
Mészáros, 2001; Sari & Esin, 2001; Nakar et al., 2009; Petropoulou &
Mastichiadis, 2009) have explored the dynamics and photon emission (including
inverse Compton scattering) of this blast wave. They parameterise multiple
assumptions on the magnetic field in this blast wave, the particle
acceleration mechanism, and the geometry of the zone using phenomenological
parameterizations providing a second, SSC component.

We reduce this phenomenological description further to a level of simplicity
needed to answer our question. We therefore merge degeneracies in the overall
normalisation into one phenomenological normalisation parameter of the
synchrotron photon flux and consider each time bin independently.

Coherently with the fireball model, we define the blast wave as the region
behind the relativistic forward shock, which forms between the homogeneous,
isotropic plasma shell and a circum-burst medium with constant density,
$n_{\text{up}}$. We note the weak dependence of the Lorentz factor on the
upstream density $n_{\text{up}}$ (see eq. 1), even for non-constant density
profiles. When the shell is in its self-similar deceleration phase (Blandford
& McKee, 1976), one can estimate the Lorentz factor at the observation time,
$t_{\text{obs}}$, from the conservation of the total isotropic energy
$E_{\text{iso}}$ as ($m_{p}$ is the proton mass for a hydrogen dominated
circum-burst medium):

$\displaystyle\Gamma$
$\displaystyle=\left(\frac{4\pi}{3m_{\text{p}}c^{5}8^{3}}\right)^{-1/8}\left(\frac{E_{\text{iso}}}{n_{\text{up}}t_{\text{obs}}^{3}}\right)^{1/8}$
(1) $\displaystyle=89\left[\left(\frac{E_{\text{iso}}}{3.5\times
10^{53}~{}{\rm erg}}\right)\left(\frac{1~{}{\rm
cm}^{-3}}{n_{\text{up}}}\right)\quantity(\frac{100~{}{\rm
s}}{t_{\text{obs}}})^{3}\right]^{1/8}.$ (2)

We note that $E_{\text{iso}}$ is estimated from the observed gamma-ray flux
and a conversion efficiency of kinetic energy to photons needs to be included,
which we take here to be 1. We used a value of $E_{\text{iso}}=3.5\times
10^{53}$ erg, $n_{\text{up}}=1\,\mathrm{cm}^{-3}$ and the geometric means
$t_{\text{obs}}=86$s and $t_{\text{obs}}=140$s. We also note that the forward
shock is moving slightly faster than the down stream material:
$\Gamma_{\text{shock}}=\sqrt{2}\Gamma$.

In order to parameterise the pressures in the downstream region of the shock,
we assume that in its rest frame the shock converts a fraction of the incoming
ram pressure $\beta^{2}\Gamma_{\text{shock}}^{2}n_{\text{up}}m_{p}c^{2}$ into
thermal pressure ($\varepsilon_{\text{th}}$), outgoing ram pressure
($\varepsilon_{\mathrm{ram}}$), non-thermal electrons ($\varepsilon_{e}$) and
turbulent magnetic fields ($\varepsilon_{\text{B}}$). The two relative
fractions, $\varepsilon_{e}$ and $\varepsilon_{\mathrm{B}}$, are assumed to be
small enough not to affect the hydrodynamics of the shock. As common within
the standard fireball model, the thermal fraction of electrons is neglected in
our model (although see e.g. Warren et al., 2022).

From the fraction $\varepsilon_{\text{B}}$ of the incoming ram pressure we
estimate the strength $B$ of the homogeneous, turbulent magnetic field as333We
adopted a factor of 32 (fraction of energy density) instead of 16 (fraction of
ram pressure) for easier comparability to the typical parameterisation in the
literature.:

$\displaystyle
B=\sqrt{32\pi\varepsilon_{\text{B}}n_{\text{up}}m_{p}c^{2}}\Gamma=0.6~{}{\rm
G}\left[\left(\frac{\varepsilon_{\text{B}}}{10^{-3.5}}\right)\left(\frac{n_{\text{up}}}{1~{}{\rm
cm}^{-3}}\right)\right]^{1/2}\frac{\Gamma}{89}.$ (3)

We emphasise that this treatment leaves the magnetic field strength as a free
fit parameter, bound by a hard upper limit at ${\varepsilon_{B}<1}$. However,
it should be noted that values of $\varepsilon_{B}$ above a few percent
already challenge the shock description.

For a given magnetic field strength and system size, the acceleration and
cooling timescales can be estimated:

$\displaystyle\tau_{\text{adi}}=\frac{2r}{3\Gamma c}=\frac{16}{3}\Gamma
t_{\text{obs}}=4.8\times
10^{4}~{}\mathrm{s}~{}\left(\frac{\Gamma}{89}\right)\left(\frac{t_{\text{obs}}}{100~{}\mathrm{s}}\right),$
(4)
$\displaystyle\tau_{\text{syn}}(E_{e})=\frac{9}{8\pi}\frac{h}{\alpha}\quantity(\frac{B_{\mathrm{c}}}{B})^{2}\frac{1}{E_{e}}=10^{3}~{}{\rm
s}\left(\frac{0.6~{}{\rm G}}{\rm
B}\right)^{2}\left(\frac{1\mathrm{TeV}}{E_{e}}\right),$ (5)
$\displaystyle\tau_{\text{acc}}(E_{e})=\eta\frac{E_{e}}{eBc}=14~{}{\rm
s}\>\eta\left(\frac{E_{e}}{76\mathrm{TeV}}\right)\left(\frac{0.6~{}{\rm
G}}{\rm B}\right).$ (6)

Here we used the critical magnetic field strength
$B_{\text{c}}=m_{e}^{2}c^{3}/\hbar e=4.4\times 10^{13}$ G, Planck’s and fine
structure constant $h=2\pi\hbar$ and $\alpha_{\text{f}}$, and the electron
rest mass energy $m_{\text{e}}c^{2}$. As can be seen from fig. 4, one can
neglect the characteristic timescale of inverse Compton scattering, given that
it never dominates compared the other timescales. We explain this in further
detail in appendix A.

We assume that the shock continuously, homogeneously, and isotropically
injects non-thermal electrons with a power law spectrum $Q\equiv\differential
N/\differential E_{e}\differential t\propto
E_{e}^{-p}\exp\quantity(-E_{e}/E_{e}^{\mathrm{max}})$, above the energy scale
$E_{e}^{\mathrm{min}}$. The maximum energy scale, $E_{e}^{\mathrm{max}}$ is
obtained by equating the acceleration and cooling timescales (eq. 8). We
emphasise that the injection process is poorly understood, and that the
commonly used descriptions (e.g. Sari et al., 1996) via time-independent
fractions for both the injected particle rate and the injected power into
these particles are ad-hoc assumptions. We fix the minimum injected energy to
$E_{e}^{\mathrm{min}}=10^{9}$ eV.

Equating $\tau_{\text{adi}}=\tau_{\text{syn}}(E_{e}^{\mathrm{cb}})$ and
$\tau_{\text{syn}}(E_{e}^{\mathrm{max}})=\tau_{\text{acc}}(E_{e}^{\mathrm{max}})$
yields the two additional energy scales:

$\displaystyle E_{e}^{\mathrm{cb}}=$
$\displaystyle\frac{9}{4\alpha_{\text{f}}}\quantity(\frac{B_{\mathrm{c}}}{B})^{2}\frac{\hbar}{\tau_{\text{adi}}}=22~{}\mathrm{GeV}\quantity(\frac{0.6~{}\mathrm{G}}{B})^{2}\quantity(\frac{4.8\times
10^{4}~{}\mathrm{s}}{\tau_{\text{adi}}}),$ (7) $\displaystyle
E_{e}^{\mathrm{max}}=$
$\displaystyle\quantity(\frac{9}{4\alpha_{\text{f}}}\frac{1}{\eta}\frac{B_{\text{c}}}{B})^{1/2}m_{\text{e}}c^{2}=76~{}\mathrm{TeV}\quantity(\frac{0.6~{}\mathrm{G}}{B})^{1/2}\quantity(\frac{1}{\eta})^{1/2}.$
(8)

The hierarchy of these three energy scales dictates specific scenarios. The
case $E_{e}^{\mathrm{min}}\leq E_{e}^{\mathrm{cb}}$
($E_{e}^{\mathrm{min}}>E_{e}^{\mathrm{cb}}$) is referred to as the slow (fast)
cooling regime (e.g. Sari et al., 1998).

Additionally, we assume that the electron spectrum quickly converges to its
time-dependent quasi-steady state distribution (as e.g. in Huang et al.,
2022), $\differential N/\differential E_{e}(t)\propto Q(t)\cdot\tau(t)$, which
holds robustly when $\tau$, the shortest timescale, is much smaller than the
timescales at which $Q$ and $\tau$ themselves change. Even for ${\tau\sim
Q/\dot{Q}\sim\tau/\dot{\tau}}$, the quasi-steady state differs from $Q\tau$
only by a constant factor of order unity. We neglect the corresponding shift
of order unity of $E_{e}^{\mathrm{cb}}$ in the slow cooling regime.

This spectrum can be phenomenologically described by a smoothly broken power
law:

$\derivative{N}{E_{e}}=\evaluated{\derivative{N}{E_{e}}}_{E_{e0}}\quantity(\frac{E_{e}}{E_{e0}})^{-p_{1}}\left[1+\quantity(\frac{E_{e}}{E_{e1}})^{s_{\text{e}}}\right]^{\frac{(p_{1}-p_{2})}{s_{\text{e}}}}e^{-\frac{E_{e}}{E_{e}^{\mathrm{max}}}}\theta(E_{e}>E_{e0}).$
(9)

For the slow cooling regime, $E_{e0}=E_{e}^{\mathrm{min}}$,
$E_{e1}=E_{e}^{\mathrm{cb}}$, $p_{1}=p$, and $p_{2}=p+1$. On the other hand,
for the fast cooling regime, $E_{e0}=E_{e}^{\mathrm{cb}}$,
$E_{e1}=E_{e}^{\mathrm{min}}$, $p_{1}=2$, and $p_{2}=p+1$. Due to the
preference of the smoothness parameter $s_{\text{e}}>10$ found in Ajello et
al. (2020), we use a value of $s_{\text{e}}=20$, which effectively yields a
broken power law for the electron spectrum.

We reduce all the complexity of the normalisation of the two components of the
photon spectrum to two parameters: 1) the normalisation of the synchrotron
component $F_{\text{syn}}$ at an observed energy of 100 keV and 2) the
relative factor of the inverse Compton component $N_{\text{IC}}$,
phenomenologically defined as the ratio of the maximum value of the energy
flux spectra of both components. We emphasise the flexibility of this approach
to model a standard one-zone SSC spectrum as well as different scenarios
leading to a single extended synchrotron component above the so called
synchrotron burn-off limit of about 100 MeV in the emission frame.

This source spectrum is then transformed to the observer’s frame by assuming a
point-like emission zone, an observation angle of $0^{\circ}$ and a redshift
of $z=0.43$. Since we observe a strong suppression of the photon flux below a
few keV we include a significant photoelectric absorption at the source and
leave the column density NH as a free parameter in our fit. Additionally, a
local absorption in our Milky Way is included ($N_{H}=7.54\times
10^{19}\mathrm{cm}^{-2}$, Ajello et al., 2020). We assume throughout that
internal $\gamma\gamma$-absorption or secondary cascading lead to only
secondary effects, and are therefore safely negligible.

###  2.2 Data reduction

GRB 190114C triggered the Fermi Gamma-Ray Burst Monitor (Fermi-GBM) at
20:57:02.63 UT ($T_{0,\mathrm{GBM}}$) and 0.56s later the Swift Burst Alert
Telescope (Swift-BAT) ($T_{0,\mathrm{BAT}}$) (Ajello et al., 2020).

Its afterglow emission ($\gtrsim$ 30s) was detected by many follow-up
observations, in particular the Swift X-ray Telescope (Swift-XRT), the Fermi
Large Area Telescope (Fermi-LAT) and the Major Atmospheric Gamma Imaging
Cherenkov (MAGIC) telescopes (MAGIC Collaboration et al., 2019b), providing an
unprecedentedly broad dataset from keV to TeV energies.

We revisit two emission phases ($67.71-110$s and $110-180$s after
$T_{0,\mathrm{BAT}}$) and perform the joint-fit spectral analysis with multi-
wavelength data: Swift-XRT, Swift-BAT, Fermi-GBM, Fermi-LAT, and MAGIC.

As a departure point to explore the stability of the model fit comparison, we
adopt a similar analysis for the Swift and Fermi detectors as was used in
Ajello et al. (2020) (our default case) and refer to the appendix B for more
details, including the few Fermi-LAT photons detected (Fig. 5). Note that this
includes a floating-normalization factor ($<\pm$ 15%, see Ajello et al., 2020)
to Swift-BAT due to the overlapping energy range with Fermi-GBM.

For the stability analysis we additionally define a floating norm 15% setup,
where we include for each detector a floating norm limited to $\pm 15\%$ to
account for the uncertainty in the cross-calibration between instruments
(Madsen et al., 2017; Tsujimoto et al., 2011; Krimm et al., 2013; Bissaldi et
al., 2011; Fermi-LAT collaboration, 6 13; Ahnen et al., 2017).

For the MAGIC data, we do not have access to the raw data, preventing any
studies of systematic effects. Instead, we use the publicly available energy
flux points in MAGIC Collaboration et al. (2019b) and interpret the
uncertainties as Gaussian standard deviations.

###  2.3 Joint-fit spectral analysis procedure

We fit the model (see Section 2.1) to the data at the counts level by forward-
folding the model flux through the instrument response and by using the proper
statistical treatment of signal and background to calculate the likelihood for
each instrument: C-stat444Poisson data with Poisson background (Cash, 1979)
for Swift-XRT, $\chi^{2}$ for Swift-BAT and MAGIC, and PG-stat555Poisson data
with Gaussian background for Fermi-GBM and Fermi-LAT (see appendix B for
details). Fig. 1 visualises the unabsorbed, best fit in energy flux space
(top) for the default SSC case with the counts (rate) of observation,
background and expectation and residual for each included detector.

Following a Bayesian approach, we derive the posterior probabilities of the
parameters and integrate the evidence $Z$ for each model (see appendix C for
details). We perform quantitative model comparison via the Bayes factor
$B_{12}=Z_{1}/Z_{2}$, stating the preference of model 1 over model 2. The
interpretation of the Bayes factor requires a scale, and we emphasize the
potential bias from this choice. A common choice is the Jeffrey’s scale, which
interprets values for $\log_{10}B_{12}$ of 1 and 2 as thresholds for moderate
and strong preference, respectively (Trotta, 2008). It should be used
carefully since the idea of unifying the interpretation of significance across
different branches of science can also be seen as a limitation of its
applicability.

Figure 1: Visualisation of our method. Above: unabsorbed energy flux with the
1-$\sigma$ envelope with its two (red and blue) components. Below: Measured
and forward folded model count rates and corresponding residuals for each
detector.

##  3 Results

We show in Fig. 2 the uncertainty bands of the energy flux, $EF_{E}$, for the
time bins (67.71-110 s and 110-180 s after $T_{0,\mathrm{BAT}}$) for the two
models (SSC and extended synchrotron only model). As appreciated from the
figure, the data require that all four model envelopes are extremely flat,
$|\Delta\log_{10}(EF_{E})/\Delta\log_{10}E|<0.5/9$. Fig. 2 indicates that in
the second time bin both models are consistent within their uncertainties,
whilst for the first time bin the SSC model differs from the synchrotron only
model by more than 1$\sigma$. However, only a statistical test, such as the
Bayesian model comparison introduced in Section 2.3, allows for any further
statement about a preference.

For our default fit case, we find that for the first time bin, a Bayes factor,
$\log_{10}(Z_{\rm SSC}/Z_{\rm syn})$, of $4$ is obtained. This indicates a
preference for the SSC model over the synchrotron only model (see Fig. 3). We
further find that in the second time bin this Bayes factor is essentially $0$.
We interpret these results as strong indication for the preference of the SSC
over the synchrotron model in the first time bin, and no preference between
the models in the second time bin.

To investigate the stability of this result, we define 4 perturbations to the
default fit configuration: (1) LAT time shift $-5\%$, where we shifted the
time selection window for Fermi-LAT photons by $-5\%$ (corresponding to 2.1
s), (2) without LAT, where we excluded the Fermi-LAT data, (3) floating norms
$15\%$, where we allowed for an effective area correction (floating norm to
account for cross-calibration uncertainties between the different detectors)
of up to $\pm 15\%$ for each instrument and (4) without XRT, where we excluded
Swift-XRT. A summary of our findings from these perturbation studies is shown
in Fig. 3.

The first two cases (1 and 2) probe the contribution of Fermi-LAT data to the
fit. The shift of the time bin window (1) is motivated by the particular
distribution of photons around the edges of the first time bin window: one
photon 0.57 s before the start of the first time bin, and two photons 0.1 s
before the end of the first time bin (see Fig. 5 in the appendix). However, we
find that the time shift (case (1)) has no significant effect on the
preference for a new component in both time bins. Since even excluding the
Fermi-LAT data completely does not change the preference, we conclude that the
few Fermi-LAT photons bring no strong contribution to the model comparison.

In cases (3) and (4) we find that either a small systematic uncertainty in the
flux or the exclusion of the Swift-XRT data removes the significant preference
for the SSC model in the first time bin ($\log_{10}(Z_{\rm SSC}/Z_{\rm
syn})<1$, see Fig. 3). In the second time bin no preference is found. In our
interpretation the preference for the SSC model in the first time bin is
driven by an increased Swift-XRT flux compared to Swift-BAT and Fermi-GBM.
Over 9 orders of magnitude in energy this small softening in the Swift-
BAT–Fermi-GBM regime would grow to a significant flux suppression at the
highest energies, which in turn would demand a new component in the VHE
regime.

Figure 2: Spectral energy distributions (SEDs) for two time bins for the two
models (68% envelopes with best fit lines) for the default case (see text).
The dashed and dotted lines show the best fit components of the SSC model. The
grey boxes on the bottom show the energy ranges of the different instruments,
which are included in the fit. Note that Fermi-GBM consists of NaI and BGO
detectors (see appendix B for details). Figure 3: Bayes factor for new
component for the default fit configuration and 4 perturbed cases (see text
for details) for the time bins 67.71-110s and 110s-180s.

##  4 Discussion and conclusions

Prior to 2018, the most constraining spectral data for probing the onset of a
new spectral component came from Swift-XRT, Nuclear Spectroscopic Telescope
Array (NuSTAR), and Fermi-LAT, observations of GRB 130427A (Kouveliotou et
al., 2013; Ackermann et al., 2014). This GRB was the brightest afterglow that
Fermi-LAT had observed in the proceeding decade since its launch in 2008. No
evidence for the onset of a new spectral component was found for this GRB (and
for others). Post 2018, the VHE gamma-ray detections of GRB afterglow emission
now allow the nature of this emission to be critically addressed.

Observationally, for all VHE detected GRB, the apparent similarity in the
level of the energy flux from the X-ray to VHE band over a broad range of
energies and time periods is striking (H.E.S.S. Collaboration et al., 2019;
MAGIC Collaboration et al., 2019b; H.E.S.S. Collaboration et al., 2021).
Although on energetic grounds it seems challenging for this emission to be
synchrotron in origin, such a model can be realised via a multi-zone
description, which itself seems a natural extension of the standard model to
consider (Khangulyan et al., 2021).

Motivated to look deeper into broadband multi-wavelength analysis for GRB
190114C, we carried out a counts level fit of both synchrotron and SSC models,
for both the first (67.71-110s) and second (110-180s) time bin in the early
afterglow phase of the GRB. Our default analysis results for GRB 190114C, not
taking into account systematics, do indeed indicate a preference for the SSC
description over the (single component) synchrotron description for the first
time bin (67.71-110s), consistent with the results reported in MAGIC
Collaboration et al. (2019b). However, no such preference is found for the
second time bin (110-180s). Furthermore, we find that this preference for one
model in the first time bin data is not stable. Our stability analysis of the
strength of preference for the SSC model over the synchrotron model finds that
such a preference disappears once systematics between instruments are taken
into account. Additionally, we find that the Fermi-LAT data set does not play
a relevant role in this model comparison. Instead, the normalisation of the
Swift-XRT data set is driving the preference. It is worth mentioning that we
cannot exclude the possibility of the decaying X-ray prompt component still
contributing in the first time bin (see fig. 2 in Ajello et al. (2020) and
fig.5 in Ursi et al. (2020)).

We conclude that no stable preference for either the SSC model or the
synchrotron only model can be drawn from the data in the two time bins. We
emphasize that this result is still compatible with the H.E.S.S. findings for
GRB 190829A (Huang et al., 2022).

We expect these results to be general, given that GRB 190114C does not appear
exceptional in terms of spectral shape. At energies above the Swift-XRT energy
range, a photon index of $\sim 2$ was consistently observed by all instruments
(see Table 1 of the appendix), in agreement with the extended-time-window
ensemble index seen by Fermi-LAT (Ajello et al., 2019). Within the Swift-XRT
energy range, a harder photon index of around $\sim 1.6$ was observed, also
consistent with the ensemble average afterglow index for Swift-XRT for the
subset jointly detected by Fermi-LAT (Ajello et al., 2018).

Our results here motivate that similar investigations be carried out in the
future for other bright and local GRBs with good multi-wavelength datasets. In
particular, a large number of such GRBs are anticipated to be detected by the
next-generation VHE telescope, the Cherenkov Telescope Array (CTA).

## Acknowledgements

The authors acknowledge support from DESY (Zeuthen, Germany), a member of the
Helmholtz Association HGF. The authors would also like to thank Phil Evans,
Andy Beardmore, and Kim Page for helpful input on the Swift-XRT. This work was
supported by the International Helmholtz-Weizmann Research School for
Multimessenger Astronomy, largely funded through the Initiative and Networking
Fund of the Helmholtz Association.

## Data Availability

This analysis used the publicly available data from instruments on board of
the Swift and Fermi satellites, which has been processed with the publicly
available software developed by the corresponding collaborations. Additionally
the data points from MAGIC Collaboration et al. (2019b) have been used. The
fits have been performed with the publicly available Multi-Mission Maximum
Likelihood framework (3ML; Vianello et al., 2015) and the UltraNest package
(Buchner, 2021). The code of the authors will be made available upon request.

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##  Appendix A Electron Broken Power-Law Description

Following Jones (1965) we here demonstrate the negligible impact of inverse
Compton scattering on the cooled electron spectrum. We adopt here a
representative set of parameters: ${t_{\mathrm{obs}}=100~{}\mathrm{s}}$,
${E_{\text{iso}}=3.5\times 10^{53}~{}\mathrm{erg}}$, ${n_{\text{up}}=1{\rm
cm}^{-3}}$, ${N_{\rm IC}=1}$, ${\varepsilon_{B}=10^{-3.5}}$, ${\eta=1}$,
${p=2}$, ${E_{e}^{\mathrm{min}}=10^{9}~{}\mathrm{eV}}$. This set of parameters
fixes $\varepsilon_{e}$ to a value of $\varepsilon_{e}\approx 10^{-1.5}$.

The upper panel of fig. 4 compares the obtained inverse Compton cooling
timescale $\tau_{\mathrm{IC}}$ (red) with the other relevant cooling
timescales for the electrons (see equations 4, 5 and 6). $\tau_{\mathrm{IC}}$
reflects an inversion of the synchrotron photon population (see lower panel of
fig. 4), taking into account the Klein-Nishina suppression. As can be seen
from the upper panel, the role of inverse Compton cooling can be safely
neglected, justifying the smooth broken power-law description.

The lower panel of fig. 4 shows the subsequent (comoving) spectral energy
distribution produced for the example case we consider here. As expected, the
inverse Compton peak sits at the same energy flux level as the synchrotron
peak. Indeed, by fixing the $N_{\rm IC}$ parameter to $N_{\rm IC}=1$, the
meeting of the $\tau_{\rm IC}$ and $\tau_{\rm syn}$ curves is forced to occur
at approximately the break energy scale (dictated by the energy when
$\tau_{\rm adi}=\tau_{\rm syn}$) in the upper panel of fig. 4.

Additionally, it should be emphasised that only a small fraction of the
upstream ram pressure is expected to be converted into non-thermal electrons
at the shock. Therefore values $\varepsilon_{e}\approx 1$ are incompatible
with this assumption, and would lead to a radical change to the shock jump
dynamics. We furthermore remark, that for this case of
$\varepsilon_{e}=10^{-1.5}$ the maximum inverse Compton flux is already
similar to the synchrotron one, such that this situation represents
$N_{\mathrm{IC}}=1$ motivated by the observations.

Figure 4: Above: Electron cooling timescales as a function of electron energy.
Below: comoving photon energy spectrum produced through electron energy loss
processes. See appendix A and section 2.1 for definitions of curves.

##  Appendix B Details on data reduction

For the _Swift_ data (Swift-BAT and Swift-XRT), we adopt the same data
selection as used in Ajello et al. (2020). Relevant in the Swift-XRT energy
range ($\lesssim$ 10 keV), we considered galactic (“TBabs”) and extragalactic
(“zTBabs”) absorption models, where the abundance and cross-section are set to
“wilm” (Wilms et al., 2000) and “vern” (Verner et al., 1996), respectively.
Since Swift-XRT is an imaging photon counter, an on-source and an off-source
(background) region of the image is used to determine the on-/off-source
counts, which both follow Poissonian statistics and are included using the
C-stat (Cash, 1979). Swift-BAT uses a coded mask to fit the photon counts and
we include this fit result with its Gaussian uncertainty with the standard
$\chi^{2}$ (also called S-statistic).

For Fermi-GBM, we used the data from two NaIs (n4 and n7; 52 keV to 875 keV)
and one BGO (b0; 288 keV to 36.3 MeV). Note that all NaI channels below 50 keV
are excluded due to the improper instrument response (Ajello et al., 2020).
The background for each Fermi-GBM detector is estimated from a first-order
polynomial fit before and after GRB190114C (-135s to -35s, 260s to 375s),
which shows $\chi^{2}\lesssim 1$ in all energy channels. The signal (on-
source) follows thus Poisson statistics, while the fitted background is
included using Gaussian statistics (PG-stat).

For Fermi-LAT data, we select P8R3Transient020E class events from 100 MeV to
10 GeV and then apply cuts with the region of interest (RoI) of 10∘ radius
centered on GRB190114C and the maximum zenith angle of 110∘. Fig. 5 shows the
Fermi-LAT–detected events passing the selections. For the galactic and
isotropic diffuse backgrounds, we used the latest templates provided by the
Fermi-LAT collaboration, gll_iem_v07 and iso_P8R3_TRANSIENT020E_V3_v1,
respectively.

Figure 5: Fermi-LAT photons after selection criteria as discussed in the text.
The complex variability of the prompt phase ends at around 25s (MAGIC
Collaboration et al., 2019b) and after 180s Fermi-LAT field of view is
occulted by the earth. The blue shaded region from 67.71 to 110s and the
orange one from 110 to 180s are the two time intervals with overlapping data
from all 5 instruments included in this work. Note the peculiar time binning
at the edges of the blue bin: a 1.28 GeV photon just about 0.6s and a 10.5 GeV
photon 2.4s before the first time interval, as well as two photons of roughly
0.3 GeV within around 0.1s before the transition to the second time bin.

Before performing the joint-fit spectral analysis, we fit the individual data
and cross-check with Ajello et al. (2020). The results agree within 1$\sigma$
level, see Table 1.

dataset | time bin | XSPEC | 3ML
---|---|---|---
Swift-XRT | 1 | $-1.649^{+0.068}_{-0.069}$ | $-1.75\pm 0.07$
2 | $-1.611^{+0.057}_{-0.058}$ | $-1.68\pm 0.06$
Swift-BAT | 1 | $-2.009^{+0.036}_{-0.036}$ | $-2.009\pm 0.027$
2 | $-2.021^{+0.040}_{-0.041}$ | $-2.001\pm 0.032$
Fermi-GBM | 1 | $-1.989^{+0.093}_{-0.101}$ | $-1.94\pm 0.10$
2 | $-1.822^{+0.118}_{-0.138}$ | $-1.81\pm 0.14$
Swift-BAT+Fermi-GBM | 1 | $-2.007^{+0.034}_{-0.034}$ | $-2.004\pm 0.027$
2 | $-2.007^{+0.039}_{-0.039}$ | $-1.995\pm 0.032$
Swift-BAT+Fermi-GBM+Fermi-LAT | 1 | $-2.027^{+0.027}_{-0.028}$ | $-2.036\pm 0.023$
2 | $-2.017^{+0.030}_{-0.031}$ | $-2.013\pm 0.026$
Table 1: Validation of 3ML results against XSPEC: We compare the spectral
index of a single power law fit for time bins 1 (67.71-110s after
$T_{0,\mathrm{BAT}}$) and 2 (110-180s after $T_{0,\mathrm{BAT}}$). Results
agree within $1\sigma$-level.

##  Appendix C Details on joint fitting procedure and model comparison

We utilize a python package, the Multi-Mission Maximum Likelihood framework
(3ML; Vianello et al., 2015), for our analysis.

We define $\mathcal{L}(\mathcal{D}|\vec{\theta},\mathcal{M})$ as the
probability to obtain a sample of data ($\mathcal{D}$) conditioning for a set
of parameters ($\vec{\theta}$) and explicitly also the model ($\mathcal{M}$).
Following a Bayesian approach, we define $\pi(\vec{\theta})$ as the prior
probability distribution of $\vec{\theta}$. We assume uniform
($p\in[1.5,2.5]$) and log-uniform ($\varepsilon_{B}\in[10^{-5},1]$,
$\eta\in[10^{-3},10^{5}]$, $F_{\text{syn}}\in[10^{-9},10^{-7}]$ and
$N_{\text{IC}}\in[10^{-2},10]$) priors. We derive the posterior probability
distribution and the evidence $Z$ (e.g. Kass & Raftery, 1995; Trotta, 2008)

$\displaystyle
Z=\int\differential\vec{\theta}\;\mathcal{L}(\mathcal{D}|\vec{\theta},\mathcal{M})\>\pi(\vec{\theta},\mathcal{M})\>,$
(10)

with the nested sampling Monte Carlo algorithm MLFriends, implemented in the
python package UltraNest (Buchner, 2021). $Z$ intuitively represents the
posterior averaged over the entire parameter space. It is of particular
interest for our case of model comparison, since it allows for a quantitative
measure of our preference of model $\mathcal{M}_{1}$ over $\mathcal{M}_{2}$
via the Bayes factor $B_{12}=Z_{1}/Z_{2}$.

In Figs. 6, 7, 8 and 9 we show the posterior distributions obtained for the
default case for both time bins.

Figure 6: Posterior distributions for SSC default fit for the first time
interval $67.71-110$s. NH is the column density of the photoelectric
absorption in $10^{22}$ atoms per cm2, $p$ is the spectral index of the
uncooled electrons, $\varepsilon_{B}=10^{\mathrm{lgepsB}}$ defines the
magnetic field strength, $\eta=10^{\mathrm{lgeta}}$ is inversely proportional
to the maximum energy of the electrons and thus also photons,
$N_{\text{IC}}=10^{\mathrm{lgN}}$ is the relative height of the inverse
Compton component to the syncchrotron component and
$F_{\text{syn}}=10^{\mathrm{lgNorm}}$ is the reference value for the
synchrotron photon flux in $erg/(cm^{2}s)$ at $E_{\gamma}^{\text{obs}}=100$
keV. Figure 7: Posterior distributions for SSC default fit for the second time
interval $110-180$s. Notation as in Fig. 6 Figure 8: Posterior distributions
for syn. only ($N_{\text{IC}}=0$) default fit for the first time interval
$67.71-110$s. Notation as in Fig. 6 Figure 9: Posterior distributions for syn.
only ($N_{\text{IC}}=0$) default fit for the second time interval $110-180$s.
Notation as in Fig. 6


# Diagonal Ramsey numbers in multipartite graphs related to stars

Chula Jayawardene
Department of Mathematics
University of Colombo, Sri Lanka.
<EMAIL_ADDRESS>

###### Abstract

Abstract: Let the star on $n$ vertices, namely $K_{1,n-1}$ be denoted by
$S_{n}$. If every two coloring of the edges of a complete balanced
multipartite graph $K_{j\times s}$ there is a copy of $S_{n}$ in the first
color or a copy of $S_{m}$ in the second color, then we will say $K_{j\times
s}\rightarrow(S_{n},S_{m})$. The size Ramsey multipartite number
$m_{j}(S_{n},S_{m})$ is the smallest natural number $s$ such that $K_{j\times
s}\rightarrow(S_{n},S_{m})$. In this paper, we obtain the exact values of the
size Ramsey numbers $m_{j}(S_{n},S_{m})$ for $n,m\geqslant 3$ and $j\geqslant
3$.

##  1 Introduction

In this paper we concentrate on simple graphs without loops and multiple
edges. Let the complete multipartite graph having $j$ uniform sets of size $s$
be denoted by $K_{j\times s}$ and the complete bipartite graph on $n+m$
vertices be denoted by $K_{n,m}$. Given, three graphs $K_{N}$, $G$ and $H$, we
say that $K_{N}\rightarrow(G,H)$ if $K_{N}$ is colored by two colors red and
blue and it contains a copy of G(in the first color) or a copy of H(in the
second color). Using this notation we define the classical Ramsey number
$r(n,m)$ as the smallest integer $N$ such that
$K_{N}\rightarrow(K_{n},K_{m})$. Sadly to say, even in the case of diagonal
classical Ramsey numbers $r(n,n)$ almost nothing significant is known beyond
the case $n=5$ (see Radziszowski et al 2014 for a survey).

In the decades that followed there are several interesting variations that
have originated from these classical Ramsey number. One obvious variation is
the case of size Ramsey numbers build up mainly by Erdös, Faudree, Rousseau
and Shelph (see Erdös et al 1978 and Faudree et al 1975). Another variation
introduced recently by Buger et al 2004 and Syafrizal et al 2005, is the
concept of balanced multipartite Ramsey numbers. This concept is based on
exploring the two colorings of multipartite graphs $K_{j\times s}$ instead of
the complete graph. Formally define size Ramsey multipartite number
$m_{j}(G,H)$ as the smallest natural number $s$ such that $K_{j\times
s}\rightarrow(G,H)$. However, sad to say currently there are very few known
balanced multipartite Ramsey numbers between pairs of graphs and pairs of
classes of graphs other than the ones introduced initially by Syafrizal et al
2005 and 2009).

## Notation

Given a graph $G=(V,E)$ with the order of the graph is denoted by $|V(G)|$ and
the size of the graph is denoted by $|E(G)|$. For a vertex $v$ of a graph $G$,
the neighborhood of $v$ is denoted by $N(v)$ and is defined as the set of
vertices adjacent to $v$. Further the cardinality of this set, denoted $d(v)$,
is defined as the degree of $v$. We say that a graph $G$ is a $k$-regular
graph if $d(v)=k$ for all $v\in V(G)$. Given a red-blue coloring of
$K_{j\times s}=H_{R}\oplus H_{B}$. The red degree and blue degree of any
vertex $v$ belonging to $V(K_{j\times s})=\\{$$v_{k,i}$$|$ $0\leqslant
i\leqslant s-1,$ $0\leqslant k\leqslant j-1\\}$ denoted by $d_{R}(v)$ and
$d_{B}(v)$ respectively, are defined as the degree of vertex $v$ in $H_{R}$
and $H_{B}$ respectively.

Given $w\geqslant 2$, $0\leqslant i\leqslant w-1$ and $0<c\leqslant w-1$,
define $\sigma_{c,w}(i)=\\{a_{1}\\}\cup\\{a_{2}\\}$,
$\sigma^{+}_{c,w}(i)=\\{a_{1}\\}$, $\sigma^{-}_{c,w}(i)=\\{a_{2}\\}$ and
$B_{0,w}(i)=\phi$ and if $k>0$, $B_{k,w}(i)=\cup_{c=1}^{k}\sigma_{c,w}(i)$
where

$a_{1}=\begin{cases}i+c&\text{ if }i+c\leqslant w-1\\\ \hskip 20.0pt&\\\
i+c-w&\text{ if }i+c>w-1\\\ \end{cases}$ $a_{2}=\begin{cases}i-c&\text{ if
}i-c\geqslant 0\\\ \hskip 20.0pt&\\\ i-c+w&\text{ if }i-c<0\\\ \end{cases}$

##  2 Some Lemmas

In all the following lemmas assume $d>0$ as the results are trivially true
when $d=0$.

######  Lemma 1.

There exists a regular induced subgraph of degree $d$ of $K_{j\times s}$ on
the vertex set $V(K_{j\times s})$ provided that $d$ is even, $j$ is odd and
$s$ is odd.

###### Proof.

If $d=2k_{1}(j-1)+2k_{2}$ for some non negative integers $k_{1}$ and $k_{2}$
such that $2k_{1}\leqslant s-1$ and $0<2k_{2}\leqslant j-1$. Construct a graph
by connecting the vertices $v_{i,l}$ and $v_{p,r}$ if one of the following
situations hold

a) If $r\in B_{k_{1},s}(l)$ and $p\neq i$. b) If $r=l$ and $p\in
B_{k_{2},j}(i)$.

$v_{2,2}$$v_{2,4}$$v_{2,3}$$v_{1,2}$$v_{0,2}$$v_{2,1}$$v_{1,4}$$v_{1,3}$$v_{1,1}$$v_{0,1}$$v_{2,0}$$v_{0,4}$$v_{0,3}$$v_{1,0}$$v_{0,0}$

Figure 1: In the case when $d=6=2k_{1}(j-1)+2k_{2}=(2\times 1)(3-1)+2\times 1$

We know that $K_{j\times s}$ consists of $j$ partite sets of size $s$. Given
$v_{i,l}$ $0\leqslant i\leqslant j-1$, $0\leqslant l\leqslant s-1$, the set
$\\{v_{p,r}|$ $p\neq i$ and $r\in B_{k_{1},s}(l)\\}$ will represent the
vertices not belonging to the $i^{th}$ partite set (denoted by $V_{i}$) that
are at most $2k_{1}$ distance apart inside a partite set (with respect to the
second coordinate), as illustrated in the following figure.

$v_{i,l}$$v_{i,i-k}$$V_{i-1}$$v_{i,s-1}$$v_{i,0}$$v_{i,l+k}$$V_{i}$$V_{i+1}$$V_{j-1}$$V_{0}$$v_{0,l}$$v_{0,s-1}$$v_{0,0}$$v_{i-1,l}$$v_{i-1,s-1}$$v_{i-1,0}$$v_{i+1,l}$$v_{i+1,s-1}$$v_{i+1,0}$$v_{j-1,l}$$v_{j-1,s-1}$$v_{j-1,0}$$k_{1}i\>\>vertices$$(j-1-i)k_{1}\>\>vertices$$k_{1}i\>\>vertices$$(j-1-i)k_{1}\>\>vertices$

Figure 2: The set consisting of $2k_{1}(j-1)$ vertices corresponding to
part(a), namely $\\{v_{p,r}|$ $p\neq i$ and $r\in B_{k_{1},s}(l)\\}$

More precisely, it will consist of the vertices

$v_{0,l-k_{1}},...,v_{i-1,l-k_{1}},v_{i+1,l-k_{1}},...,v_{j-1,l-k_{1}}$

$v_{0,l-k_{1}+1},...,v_{i-1,l-k_{1}+1},v_{i+1,l-k_{1}+1},...,v_{j-1,l-k_{1}+1}$

…

$v_{0,l-1},...,v_{i-1,l-1},v_{i+1,l-1},...,v_{j-1,l-1}$

$v_{0,l+1},...,v_{i-1,l+1},v_{i+1,l+1},...,v_{j-1,l+1}$

…

$v_{0,l+k_{1}-1},...,v_{i-1,l+k_{1}-1},v_{i+1,l+k_{1}-1},...,v_{j-1,l+k_{1}-1}$

$v_{0,l+k_{1}},...,v_{i-1,l+k_{1}},v_{i+1,l+k_{1}},...,v_{j-1,l+k_{1}}$

That is such a set consists of $2k_{1}(j-1)$ vertices.

Similarly, given $v_{i,l}$ where $0\leqslant i\leqslant j-1$ and $0\leqslant
l\leqslant s-1$ the set $\\{v_{p,r}|$ $r\neq l$ and $p\in B_{k_{2},j}(i)\\}$
will represent the vertices not belonging to the $i^{th}$ partite set(denoted
by $V_{i}$) that are at most $2k_{2}$ distance apart between partite sets
(with respect to the first coordinate), as illustrated in the following
figure. More precisely, it will consist of the vertices

$v_{i-k_{2},l},...,v_{i-1,l},v_{i+1,l},...,v_{i+k_{2},l}$

That is such a set consists of $2k_{2}$ vertices.

$v_{i,l}$$v_{i,i-k_{1}}$$V_{i-1}$$v_{i,s-1}$$v_{i,0}$$v_{i,l+k_{1}}$$V_{i}$$V_{i+1}$$V_{j-1}$$V_{0}$$v_{0,l}$$v_{0,i-k_{1}}$$v_{0,s-1}$$v_{0,0}$$v_{0,l+k_{1}}$$v_{i-1,i-k_{1}}$$v_{i-1,s-1}$$v_{i-1,0}$$v_{i-1,l+k_{1}}$$v_{i+1,i-k_{1}}$$v_{i+1,s-1}$$v_{i+1,0}$$v_{i+1,l+k_{1}}$$v_{j-1,l}$$v_{j-1,i-k_{1}}$$v_{j-1,s-1}$$v_{j-1,0}$$v_{j-1,l+k_{1}}$$k_{2}\>\>vertices$$k_{2}\>\>vertices$

Figure 3: The set consisting of $2k_{2}$ vertices corresponding to part(b),
namely $\\{v_{p,r}|$ $r=l$ and $p\in B_{k_{2},j}(i)\\}$

Thus, by the above definition, part $(a)$ will represent $2k_{1}(j-1)$
vertices adjacent to $v_{i,l}$ belonging to
$V_{0},V_{1},V_{2},...,V_{i-1},V_{i+1},...V_{j-1}$ and part $(b)$ will
represent another $2k_{2}$ vertices adjacent to $v_{i,l}$ belonging to
$V_{i-k_{2}},$ $V_{i-k_{2}+1},...,$ $V_{i-1},V_{i+1},...,V_{i+k_{2}-1},$
$V_{i+k_{2}}$.

Therefore, the degree of $v_{i,l}$ will be equal to $2k_{1}(j-1)+2k_{2}$. Also
by the remark $l\in B_{k,w}(i)$ if and only if $i\in B_{k,w}(l)$, before the
beginning of this section, we get that the graph is well defined.

Next, if $d=(2k_{1}+1)(j-1)+2k_{2}$ for some non negative integers $k_{1}$ and
$k_{2}$ such that $2k_{1}\leqslant s-3$ and $0<2k_{2}\leqslant j-1$. Construct
a graph by connecting the vertices $v_{i,l}$ and $v_{p,r}$ if one of the
following situations hold

a) If $r\in B_{k_{1},s}(l)$ and $p\neq i$.

b) If there exists $w$ such that $4w=(j-1+2k_{2})$ and
$r\in\sigma_{k_{1}+1,s}(l)$ and $p\in B_{w,j}(i)$.

c) If there exists $w$ such that $(j-1+2k_{2})-4w=2$ and
$r\in\sigma_{k_{1}+1,s}(l)$ and $p\in B_{w,j}(i)$ or else $r=l$ and $p\in
B_{1,j}(i)$.

$v_{2,2}$$v_{1,2}$$v_{0,2}$$v_{2,1}$$v_{1,1}$$v_{0,1}$$v_{2,0}$$v_{1,0}$$v_{0,0}$

Figure 4: In the case when $d=4=(2k_{1}+1)(j-1)+2k_{2}=(2\times
0+1)\times(3-1)+2\times 1$

It should be noted that the vertex sets of part (b) and part (c) are disjoint
and that $j-1+2k_{2}$ is even as $j$ is odd. Therefore, given $v_{i,l}$, it
will be either adjacent the vertices corresponding to part (a) and part (b) or
else adjacent the vertices corresponding to part (a) and part (c) according to
whether $4w=(j-1+2k_{2})$ or else $(j-1+2k_{2})-4w=2$, respectively.

As illustrated in figure 2, the set generated by part (a) namely,
$\\{v_{p,r}|$ $p\neq i$ and $r\in B_{k_{1},s}(l)\\}$ will consist of
$2k_{1}(j-1)$ vertices.

Similarly, given $v_{i,l}$ the set generated by part (b), will represent the
vertices belonging to $V_{i-w},...,V_{i-1},V_{i+1},...V_{i+w}$ sets, that are
at most $2(k_{1}+1)$ distance apart between partite sets (with respect to the
first coordinate), as illustrated in the following figure. More precisely, it
will consist of the vertices

$v_{i-w,l-(k_{1}+1)},...,v_{i-1,l-(k_{1}+1)},v_{i+1,l-(k_{1}+1)},...v_{i+w,l-(k_{1}+1)}$

$v_{i-w,l+(k_{1}+1)},...,v_{i-1,l+(k_{1}+1)},v_{i+1,l+(k_{1}+1)},...v_{i+w,l+(k_{1}+1)}$

Such a set consists of $4w$ vertices. That is, the set consists of
$(j-1)+2k_{2}$ vertices.

$v_{i,l}$$v_{i,i-k-1}$$V_{i-w}$$v_{i,s-1}$$v_{i,0}$$v_{i,l+k+1}$$V_{i}$$V_{i+w}$$V_{j-1}$$V_{0}$$v_{0,l}$$v_{0,s-1}$$v_{0,0}$$v_{i-w,l}$$v_{i-w,s-1}$$v_{i-w,0}$$v_{i+w,l}$$v_{i+w,s-1}$$v_{i+w,0}$$v_{j-1,l}$$v_{j-1,s-1}$$v_{j-1,0}$$w\>\>vertices$$w\>\>vertices$$w\>\>vertices$$w\>\>vertices$

Figure 5: The set consisting of $4w$ vertices corresponding to part (b),
namely $\\{v_{p,r}|$ $r\in\sigma_{k_{1}+1,s}(l)$ and $p\in B_{w,j}(i)\\}$

Similarly, given $v_{i,l}$ the set generated by the later part (c), will
represent the two vertices belonging to $V_{i-1},V_{i+1}$ sets, namely
$v_{i-1,l-1}$, $v_{i+1,l+1}$. More precisely the set generated by part (c),
will consist of the vertices

$v_{i-w,l-(k_{1}-1)},...,v_{i-1,l-(k_{1}-1)},v_{i+1,l-(k_{1}-1)},...v_{i+w,l-(k_{1}-1)}$

$v_{i-1,l-1},v_{i+1,l+1}$

$v_{i-w,l-(k_{1}+1)},...,v_{i-1,l-(k_{1}+1)},v_{i+1,l-(k_{1}+1)},...v_{i+w,l-(k_{1}+1)}$

Such a set consists of $4w+2$ vertices. That is, the set consists of
$(j-1)+2k_{2}$ vertices.

$v_{i,l}$$V_{i-1}$$v_{i,s-1}$$v_{i,0}$$V_{i}$$V_{i+1}$$V_{j-1}$$V_{0}$$v_{0,l}$$v_{0,s-1}$$v_{0,0}$$v_{i-1,l}$$v_{i-1,s-1}$$v_{i-1,0}$$v_{i+1,l}$$v_{i+1,s-1}$$v_{i+1,0}$$v_{j-1,l}$$v_{j-1,s-1}$$v_{j-1,0}$$1\>vertex$$1\>vertex$

Figure 6: The set consisting of $2$ vertices corresponding to later section of
part (c), namely $\\{v_{p,r}|$ $p\in B_{w,j}(i)$ or else $r=l$ and $p\in
B_{1,j}(i)\\}$

Therefore, the degree of $v_{i,l}$ will be equal to $2k_{1}(j-1)+2k_{2}$ when
part (a)+(b) situation arrises or when part (a)+(c) situation arrises. ∎

######  Lemma 2.

There exist some uniform graphs of degree $d$ on $V(K_{j\times s})$ if $j$ is
even or $s$ is even.

###### Proof.

We approach this problem by considering the following three cases.

Case 1) If $j$ is even and $s$ is odd.

If $d=2k_{1}(j-1)+2k_{2}$ for some non negative integers $k_{1}$ and $k_{2}$
such that $2k_{1}\leqslant s-1$ and $0<2k_{2}\leqslant j-2$. Construct a graph
by connecting the vertices $v_{i,l}$ and $v_{p,r}$ if one of the following
situations hold

a) If $r\in B_{k_{1},s}(l)$ and $p\neq i$. b) If $r=l$ and $p\in
B_{k_{2},j}(i)$.

The vertex $v_{i,l}$ will be either adjacent the vertices corresponding to
part (a) or part (b) and they are respectively equal to $2k_{1}(j-1)$ and
$2k_{2}$. Therefore, we get that the degree of $v_{i,l}$ is equal to
$2k_{1}(j-1)+2k_{2}$ as required.

If $d=2k_{1}(j-1)+2k_{2}+1$ for some non negative integers $k_{1}$ and $k_{2}$
such that $2k_{1}\leqslant s-1$ and $2k_{2}\leqslant j-2$. Construct a graph
by connecting the vertices $v_{i,l}$ and $v_{p,r}$ if one of the following
situations hold

a) If $r\in B_{k_{1},s}(l)$ and $p\neq i$. b) If $r=l$ and $p\in
B_{k_{2},j}(i)$.

c) If $r=l$ and $p\in\sigma_{\frac{j}{2},j}(i)$.

$v_{0,0}$$v_{1,0}$$v_{2,0}$$v_{0,2}$$v_{1,2}$$v_{2,2}$$v_{0,1}$$v_{2,1}$$v_{3,0}$$v_{3,1}$$v_{3,2}$$Example:When$
$d=3=2\times 0\times(4-1)+2\times 1+1$$v_{1,1}$

Figure 7: In the case(1) when $d=3=2k_{1}(j-1)+2k_{2}+1=2\times
0\times(4-1)+2\times 1+1$

The vertex $v_{i,l}$ will be either adjacent the vertices corresponding to
part (a), part (b) or part (c) and they are respectively equal to
$2k_{1}(j-1)$, $2k_{2}$ and one. Therefore, we get that the degree of
$v_{i,l}$ is equal to $2k_{1}(j-1)+2k_{2}+1$ as required.

If $d=(2k_{1}+1)(j-1)+m$ for some non negative integers $k_{1}$, $k_{2}$ and
$m$ such that $2k_{1}\leqslant s-3$ and $0<m\leqslant j-1$ where $m=2k_{2}$ or
$m=2k_{2}+1$. Construct a graph by connecting the vertices $v_{i,l}$ and
$v_{p,r}$ if one of the following situations hold

a) If $r\in B_{k_{1},s}(l)$ and $p\neq i$.

b) If there exists $w$ such that $w=(j-1+m)$ div $4$,
$r\in\sigma_{k_{1}+1,s}(l)$ and $p\in B_{w,j}(i)$.

c) If there exists $w$ such that $(j-1+m)-4w=1$, $r=l$ and
$p\in\sigma_{\frac{j}{2},j}(i)$.

d) If there exists $w$ such that $(j-1+m)-4w=2$, $r=l$ and
$p\in\sigma_{1,j}(i)$.

e) If there exists $w$ such that $(j-1+m)-4w=3$, $r=l$, $p\in\sigma_{1,j}(i)$
and $p\in\sigma_{\frac{j}{2},j}(i)$ (as $j$ is even).

$v_{2,2}$$v_{1,2}$$v_{0,2}$$v_{2,1}$$v_{1,1}$$v_{0,1}$$v_{2,0}$$v_{1,0}$$v_{0,0}$$v_{3,0}$$v_{3,1}$$v_{3,2}$$v_{4,0}$$v_{4,1}$$v_{4,2}$$v_{5,0}$$v_{5,1}$$v_{5,2}$

Figure 8: In the case(1) when $d=6=(2k_{1}+1)(j-1)+2k_{2}+1=(2\times
0+1)(6-1)+2\times 0+1$

It should be noted that the vertex sets of part (b), (c), (d) and part (e) are
disjoint. Therefore, $v_{i,l}$ will be either adjacent the vertices
corresponding to part (a) and part (b) or (a) and part (c) or (a) and part (d)
or else part (a) and part (e) according to whether $4w=j-1+2k_{2}$,
$(j-1+2k_{2})-4w=1$, $(j-1+2k_{2})-4w=2$ or $(j-1+2k_{2})-4w=3$ respectively.
As done before, in all these scenarios we get $d=(2k_{1}+1)(j-1)+m$ as
required.

Case 2) If $j$ is even and $s$ is even.

If $d=2k_{1}(j-1)+2k_{2}$ for some non negative integers $k_{1}$ and $k_{2}$
such that $2k_{1}\leqslant s-2$ and $0<k_{2}\leqslant j-1$. Construct a graph
by connecting the vertices $v_{i,l}$ and $v_{p,r}$ if one of the following
situations hold

a) If $r\in B_{k_{1},s}(l)$ and $p\neq i$.

b) If $k_{2}<\frac{j}{2}$, $r=l$ and $p\in B_{k_{2},j}(i)$.

c) If $\frac{j}{2}\leqslant k_{2}<j-1$ and (($r=l$ and $p\in
B_{\frac{j-2}{2},j}(i)$) or ($r\in\sigma_{\frac{s}{2},s}(l)$ and $p\in
B_{\frac{2k_{2}-(j-2)}{2},j}(i)$)).

d) If $k_{2}=j-1$ and (($r=l$ and $p\neq i$) or
($r\in\sigma_{\frac{s}{2},s}(l)$ and $p\neq i$)).

It should be noted that the vertex sets of part (b), (c) and part (d) are
disjoint. Therefore, $v_{i,l}$ will be either adjacent the vertices
corresponding to part (a) and part (b) or (a) and part (c) or else part (a)
and part (d) according to whether $k_{2}<\frac{j}{2}$, $\frac{j}{2}\leqslant
k_{2}<j-1$ or $k_{2}=j-1$ respectively. As done before, in all these scenarios
we get $d=2k_{1}(j-1)+2k_{2}$ as required.

If $d=2k_{1}(j-1)+2k_{2}+1$ for some non negative integers $k_{1}$ and $k_{2}$
where such that $2k_{1}\leqslant s-2$ and $k_{2}<j-1$. Construct a graph by
connecting the vertices $v_{i,l}$ and $v_{p,r}$ if one of the following
situations hold

a) If $r\in B_{k_{1},s}(l)$ and $p\neq i$.

b) If $k_{2}\leqslant\frac{j-2}{2}$, $r=l$ and ($p\in B_{k_{2},j}(i)$ or
$p\in\sigma_{\frac{j}{2},j}(i)$).

c) If $k_{2}\geqslant\frac{j}{2}$ and (($r=l$ and $p\neq i$) or
($r\in\sigma_{\frac{s}{2},s}(l)$ and $p\in B_{\frac{2k_{2}-(j-2)}{2},j}(i)$)).

$v_{0,0}$$v_{1,0}$$v_{2,0}$$v_{0,2}$$v_{1,2}$$v_{2,2}$$v_{0,1}$$v_{3,1}$$v_{2,1}$$v_{0,3}$$v_{1,3}$$v_{2,3}$$v_{3,0}$$v_{1,1}$$v_{3,2}$$v_{3,3}$

Figure 9: In the case(2) when $d=5=2k_{1}(j-1)+2k_{2}+1=2\times
0\times(4-1)+2\times 2+1$

It should be noted that the vertex sets of part (b) and part (c) are disjoint.
Therefore, $v_{i,l}$ will be either adjacent the vertices corresponding to
part (a) and part (b) or (a) and part (c) according to whether
$k_{2}\leqslant\frac{j-2}{2}$ or $k_{2}\geqslant\frac{j}{2}$ respectively. As
done before, in all these scenarios we get $d=2k_{1}(j-1)+2k_{2}+1$ as
required.

Case 3) If $j$ is odd and $s$ is even.

If $d=2k_{1}(j-1)+2k_{2}$ for some non negative integers $k_{1}$ and $k_{2}$
such that $2k_{1}\leqslant s-2$ and $0<k_{2}\leqslant j-1$. Construct a graph
by connecting the vertices $v_{i,l}$ and $v_{p,r}$ if one of the following
situations hold

a) If $r\in B_{k_{1},s}(l)$ and $p\neq i$.

b) If $k_{2}<\frac{j}{2}$, $r=l$ and $p\in B_{k_{2},j}(i)$

c) If $k_{2}\geqslant\frac{j}{2}$ and (($r=l$ and $p\neq i$) or
($r\in\sigma_{\frac{s}{2},s}(l)$ and $p\in B_{\frac{2k_{2}-(j-1)}{2},j}(i)$)).

It should be noted that the vertex sets of part (b) and part (c) are disjoint.
Therefore, $v_{i,l}$ will be either adjacent the vertices corresponding to
part (a) and part (b) or (a) and part (c) according to whether
$k_{2}<\frac{j}{2}$ or $k_{2}\geqslant\frac{j}{2}$ respectively. As done
before, in all these scenarios we get $d=2k_{1}(j-1)+2k_{2}$ as required.

Case 4) If $j$ is odd and $s$ is even.

If $d=2k_{1}(j-1)+2k_{2}+1$ for some non negative integers $k_{1}$ and $k_{2}$
such that $2k_{1}\leqslant s-2$ and $0\leqslant k_{2}<j-1$. Construct a graph
by connecting the vertices $v_{i,l}$ and $v_{p,r}$ if one of the following
situations hold

a) If $r\in B_{k_{1},s}(l)$ and $p\neq i$.

b) If $k_{2}<\frac{j-1}{2}$ and (($r=l$ and $p\in B_{k_{2},j}(i)$) or
($r\in\sigma_{\frac{s}{2},s}(l)$ and $r>l$ and $p=\sigma^{+}_{k_{2}+1,j}(i)$)
or ($r\in\sigma_{\frac{s}{2},s}(l)$ and $r<l$ and
$p=\sigma^{-}_{k_{2}+1,j}(i)$)).

c) If $k_{2}=\frac{j-1}{2}$and (($r=l$ and $p\neq i$) or
($r\in\sigma_{\frac{s}{2},s}(l)$ and $r>l$ and $p=\sigma^{+}_{k_{2}+1,j}(i)$)
or ($r\in\sigma_{\frac{s}{2},s}(l)$ and $r<l$ and
$p=\sigma^{-}_{k_{2}+1,j}(i)$)).

d) If $k_{2}>\frac{j-1}{2}$ and (($r=l$ and $p\neq i$) or
($r\in\sigma_{\frac{s}{2},s}(l)$ and $p\in B_{\frac{2k_{2}-(j-1)}{2},j}(i)$)
or ($r\in\sigma_{\frac{s}{2},s}(l)$ and
$p=\sigma^{+}_{\frac{2k_{2}-(j-1)}{2}+1,j}(i)$ with $r>l$ ) or
($r\in\sigma_{\frac{s}{2},s}(l)$ and
$p=\sigma^{-}_{\frac{2k_{2}-(j-1)}{2}+1,j}(i)$ with $r<l$)).

$v_{2,2}$$v_{1,2}$$v_{0,2}$$v_{2,1}$$v_{1,1}$$v_{0,1}$$v_{2,0}$$v_{1,0}$$v_{0,0}$$v_{2,3}$$v_{0,3}$$v_{1,3}$

Figure 10: In the case(3) when $d=5=2k_{1}(j-1)+2k_{2}+1=2\times
1\times(3-1)+2\times 0+1$

It should be noted that the vertex sets of part (b), (c) and part (d) are
disjoint. Therefore, $v_{i,l}$ will be either adjacent the vertices
corresponding to part (a) and part (b) or (a) and part (c) or else (a) and
part (d) according to whether $k_{2}<\frac{j-1}{2}$, $k_{2}=\frac{j-1}{2}$ or
$k_{2}>\frac{j-1}{2}$ respectively. As done before, in all these scenarios we
get $d=2k_{1}(j-1)+2k_{2}+1$ as required. ∎

######  Lemma 3.

$m_{j}(S_{n},S_{m})\leqslant\left\lceil\dfrac{n+m-3}{j-1}\right\rceil$ where
$j,n,m\geqslant 3$.

###### Proof.

Consider any red/blue coloring given by $K_{j\times s}=H_{R}\oplus H_{B}$,
where $s=\left\lceil\dfrac{n+m-3}{j-1}\right\rceil$, such that $H_{R}$
contains no red $S_{n}$. Let $v$ be any vertex of $K_{j\times s}$. Then $v$ is
incident to at most $n-2$ red edge. Hence,

$d_{B}(v)\geqslant\left\lceil\dfrac{n+m-3}{j-1}\right\rceil(j-1)-(n-2)\geqslant
m-1$

Therefore, $H_{B}$ will contain a blue $S_{m}$. Hence the result. ∎

######  Lemma 4.

$m_{j}(S_{n},S_{m})\geqslant\left\lceil\dfrac{n+m-4}{j-1}\right\rceil$ where
$j,n,m\geqslant 3$.

###### Proof.

Consider the red and blue coloring of $K_{j\times s}$ given by $K_{j\times
s}=H_{R}\oplus H_{B}$, where $s=\left\lceil\dfrac{n+m-4}{j-1}\right\rceil-1$,
where all the vertices will have uniform red degree of $n-2$ or $n-3$ (this is
possible by lemma 2). Then clearly $H_{B}$ does not contain a red $S_{n}$. Let
$v$ be any vertex of $K_{j\times s}$. Then,

$d_{B}(v)=\left(\left\lceil\dfrac{n+m-4}{j-1}\right\rceil-1\right)(j-1)-(n-3)$

$=\left\lceil\dfrac{n+m-4}{j-1}\right\rceil(j-1)-j+1-n+3$

$\geqslant n+m-4-j-n+4\geqslant m-j$

Therefore, $H_{B}$ will not contain a blue $S_{m}$. Hence the result. ∎

######  Lemma 5.

$m_{j}(S_{n},S_{m})=\left\lceil\dfrac{n+m-4}{j-1}\right\rceil$ if $(n+m-4)\neq
0$ mod $(j-1)$ where $j,n,m\geqslant 3$.

###### Proof.

We know that if $(n+m-4)\neq 0$ mod $(j-1)$ then
$\left\lceil\dfrac{n+m-4}{j-1}\right\rceil=\left\lceil\dfrac{n+m-3}{j-1}\right\rceil$.
Hence the result follows by lemma 3 and lemma 4.

∎

######  Lemma 6.

Suppose that $j,n,m\geqslant 3$. Then,
$m_{j}(S_{n},S_{m})\leqslant\left\lceil\dfrac{n+m-4}{j-1}\right\rceil$
provided that $(n+m-4)=0$ mod $(j-1)$ with $j$ is odd, $n$ is odd and
$s=\dfrac{n+m-4}{j-1}$ is odd.

###### Proof.

Consider any red/blue coloring given by $K_{j\times s}=H_{R}\oplus H_{B}$,
where $s=\left\lceil\dfrac{n+m-4}{j-1}\right\rceil$, such that $H_{R}$
contains no red $S_{n}$. Since, $j\times s\times(n-2)$ is odd, there will
exist at least one vertex $v\in K_{j\times s}$ such it is not incident to
$n-2$ red edges, as otherwise by handshake lemma $j\times
s\times(n-2)=2|E(H_{R})|$, a contradiction. Hence,

$d_{B}(v)\geqslant\left\lceil\dfrac{n+m-4}{j-1}\right\rceil(j-1)-(n-3)\geqslant
m-1$

Therefore, $H_{B}$ will contain a blue $S_{m}$. Hence the result. ∎

######  Lemma 7.

Suppose that $j,n,m\geqslant 3$. Then,
$m_{j}(S_{n},S_{m})\geqslant\left\lceil\dfrac{n+m-3}{j-1}\right\rceil$
provided that $(n+m-4)=0$ mod $(j-1)$ with $j$ is even or
$s=\dfrac{n+m-4}{j-1}$ even or $n$ is even.

###### Proof.

By lemma 3 and lemma 4, $K_{j\times s}$ where
$s=\left\lceil\dfrac{n+m-3}{j-1}\right\rceil-1=\left\lceil\dfrac{n+m-4}{j-1}\right\rceil$,
will have a $n-2$ regular subgraph on $K_{j\times s}$. Using this subgraph
generate a red/blue coloring given by $K_{j\times s}=H_{R}\oplus H_{B}$, where
all the edges of this subgraph are colored red and all other edges colored
blue. Then clearly $H_{R}$ is $S_{n}$ \- free. Furthermore, for any vertex
$v\in K_{j\times s}$,
$d_{B}(v)=\left(\dfrac{n+m-4}{j-1}\right)(j-1)-(n-2)=m-2$. Therefore, $H_{B}$
will not contain a blue $S_{m}$. Hence the result. ∎

######  Theorem 8.

If $j\geqslant 3$ and $n,m\geqslant 2$ then,

$m_{j}(S_{n},S_{m})=\begin{cases}\left\lceil\dfrac{\max\\{n,m\\}-1}{j-1}\right\rceil&\text{
if }n=2\text{ or }m=2\\\ \hskip 14.0pt&\\\
\left\lceil\dfrac{n+m-4}{j-1}\right\rceil&\text{ if }n+m-4=(j-1)s;j,s,n\\\
&\text{ are odd and }n,m\geqslant 3\\\ \hskip 14.0pt&\\\
\left\lceil\dfrac{n+m-3}{j-1}\right\rceil&\text{ otherwise }\\\ \end{cases}$

###### Proof.

The theorem clearly follows from lemmas 5, 6 and 7 as
$m_{j}(S_{2},S_{m})=\left\lceil\dfrac{m-1}{j-1}\right\rceil$ (see Syafrizal et
al 2005). ∎

## References

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