[a,b]Peter Križan (for the Belle II Collaboration)

# The Belle II Upgrade Program

###### Abstract

The Belle II detector at the SuperKEKB accelerator complex is covering a wide
range of exciting physics topics. To achieve the project’s research goals, a
substantial increase of the data sample to 50 ab-1 is needed, and for that,
the luminosity has to reach the ambitious goal of $6\times 10^{35}$ cm-2 s-1.
The progress towards the design luminosity is accompanied by research and
development of the accelerator, detector components, operation methods, as
well as their upgrades. In the present contribution, we will discuss the
status and plans of the project, timescales for upgrades, their motivations,
and opportunities, an overview of upgrade options, and finish with an outlook
and perspectives.

##  1 Introduction

The Belle II detector [1, 2] at the SuperKEKB accelerator complex [3] is a
super-B factory covering a wide range of exciting physics topics [4]. To
achieve the project’s research goals, a substantial increase of the data
sample to 50 ab-1 is needed, and for that, the luminosity has to reach the
ambitious goal of $6\times 10^{35}$ cm-2 s-1. The progress towards the design
luminosity is accompanied by research and development of the accelerator,
detector components, operation methods, as well as their upgrades.

In what follows, we will discuss the status and plans of the project,
timescales for upgrades, their motivations, and opportunities, an overview of
upgrade options, and finish with an outlook and perspectives.

##  2 The SuperKEKB/Belle II program and plans

The first phase of SuperKEKB operation in 2016, without the detector and
collisions, was carried out to test the injection step and the two accelerator
rings and for baking the 3 km of the accelerator vacuum chambers. This was
followed by Phase 2 in 2018, with the first collisions with the complete
accelerator but with an incomplete detector. Instead of the vertex detector, a
dedicated background detector system known as Beast 2 was employed [5].

Physics runs started in Phase 3 (from 2019) luminosity runs with a complete
detector, where the vertex detector was comprised of the pixel Detector (PXD)
with a fully instrumented first layer and a partly instrumented layer 2, while
the 4-layer micro-strip detector (SVD) was fully instrumented. The first
physics paper appeared in January 2020 [6].

The new accelerator presented several challenges, with an added operational
complexity during the pandemic. It nevertheless reached a record peak
luminosity of $4.7\times 10^{34}$ cm-2 s-1. The path to reach the next
milestone, a luminosity of $2\times 10^{35}$ cm-2 s-1, has been identified.
There are, however, still significant factors to achieve the target peak
luminosity of $6\times 10^{35}$ cm-2 s-1.

Figure 1: Specific and integrated luminosity projection of SuperKEKB.

The path toward higher luminosity is a steep one. Challenges have been
encountered in machine performance and stability coming from beam blow-up due
to beam-beam effects, lower than expected beam lifetime, transverse mode
coupling instabilities, low machine stability, injector capability, and aging
infrastructure. Another set of challenges is background in the detector with
single beam sources (beam-gas and Touschek scattering), luminosity-related
(radiative Bhabha and two-photon processes), and injection background.

Mitigation measures include a consolidation of the accelerator complex and the
detector. An international task force was established in 2020 to help with
advice and ideas on how to consolidate the accelerator. Many countermeasures
are under development. To go beyond the luminosity of $\approx 2\times
10^{35}$ cm-2 s-1, a major redesign of the Interaction Region may be required.

To consolidate the detector, the complete PXD detector has to be installed.
Also, a part of the light sensors for one of the particle identification
devices, the TOP (Time-of-Propagation) detector, have to be replaced by more
robust devices. These two upgrades will be carried out in the Long Shutdown 1
(LS1) that started end of June 2022 and is expected to be finished by autumn
2023.

A further upgrade of the detector is envisaged to make it more robust against
backgrounds and improve its performance. This upgrade would be carried out in
the Long Shutdown 2 (LS2), expected to start by the end of 2026 or in 2027.
LS2 is motivated by a (still to be defined) redesign of the interaction region
(IR), with a replacement of the superconducting final quadrupoles. This is a
window of opportunity for significant detector upgrades, including a possible
replacement of the full vertex detector, the pixel (PXD) and silicon-strip
(SVD) based parts. Although large uncertainties accompany the extent of
accelerator upgrades in LS2, we must be ready and prepare technology choices
for possible detector upgrades.

On an even longer-term ($>2032$), options are discussed if a significant
luminosity increase would become possible. A data sample of $\approx 250$ ab-1
would be interesting, although it is not clear at this time how to realize
such a large increase in luminosity. A detailed study of the physics case is
needed, and technology R&D for an extreme-luminosity detector would have to
start soon.

##  3 Upgrades, main ideas, and time scale

Crucial performance challenges that impact the physics reach are tracking at
low momentum, vertex and interaction point resolution, calorimetry energy
resolution and lepton identification, trigger efficiency, $K/\pi$ separation,
and $K_{L}$ detection.

Figure 2: Upgrades of the Belle II detector, as discussed for LS2 (black text)
and LS3 (red text).

###  3.1 Vertex detector (VXD) upgrade

An upgrade of the vertex detector (VXD) is prepared to be aligned with a
significant interaction region redesign, also allowing for large safety
factors against background predictions. Possible performance improvements
include an improvement of the impact parameter and vertexing resolution, an
improved tracking performance for low transverse momentum ($p_{T}$) tracks,
and a possible contribution to the L1 trigger.

Several proposals are under study and in the R&D phase, taking advantage of
recent technology developments. A thin and fine-pitch DSSD option assumes a
140 $\mu$m thin sensor with a z-pitch of $<80$ $\mu$m, and a new ASIC for low
noise operation. Another option assumes an upgraded DEPFET sensor with a
higher radiation tolerance through higher gain and a faster read-out (few
$\mu$s) with new ASICs, and a possible read-out re-orientation with respect to
the present version of DEPFET. In the option with silicon-on-insulator (SOI)
pixels, the sensors are produced in the Lapis 200 nm process; the Dual Time
pixel sensor (DuTiP) has a pitch of 45 $\mu$m, and a 2x60 ns integration time.
Finally, a fully pixelated VXD concept is based on the CMOS-MAPS technology,
produced in the Tower 180 nm process, an extension of the TJ-MONOPIX2 to the
OBELIX sensor with a pitch of $<40$ $\mu$m and 100 ns integration time; it
assumes that the complete VXD detector is built of either all-Si modules or
ALICE-ITS-like ladders.

###  3.2 Central Drift Chamber (CDC)

For the Central Drift Chamber (CDC), an upgrade of the read-out electronics is
foreseen for LS2 with improved radiation tolerance and reduced cross-talk and
power consumption. A new ASIC is under design, and a new FPGA and optical
modules are being tested. Long-term studies are exploring options for
sustaining higher rates and backgrounds expected at higher luminosities. For
the present CDC, studies are underway with different gas mixtures. In the
longer term, two further options are under study, one in which the inner part
of the CDC is replaced by a silicon tracking device and one where a TPC
tracker replaces the main part of the CDC with a Gridpix-like pixel read-out.

###  3.3 Particle identification detectors

TOP, Time-of-Propagation detector. Here, the plan is to install life-extended
Atomic Layer Deposition (ALD) micro-channel plate (MCP) PMTs to replace the
standard MCP PMTs (in LS1), and possibly to replace the standard ALD MCP PMTs
(in LS2). A study of SiPMs as possible MCP PMT replacement is being pursued;
it would require a dedicated cooling system and is thus possible only at a
longer time scale (beyond LS2). A read-out electronics upgrade is under study
where the present IRSX 8-channel 250 $\mu$m CMOS ASIC would be replaced by a
TOPSoC 32-channel 130 $\mu$m CMOS, still under development; this new set-up
would allow for a feature extraction inside ASIC, and reduced power
consumption.

ARICH, aerogel RICH detector. The detector is performing very well and has
shown very little background susceptibility. Therefore no modifications are
planned for the LS1 and LS2. Long-term studies include a photon detector
upgrade with SiPMs or MCP-PMTs. For the SiPM option, irradiation tests are
underway. Another promising candidate, a large area MCP-PMT (LAPPD, 20
cm$\times$20 cm active area), has been tested recently, and further tests are
planned. Two options are under study for the read-out electronics of the
upgraded detector: a custom-developed version and the FASTiC chip (developed
for the next upgrade of LHCb RICHes). The impact of a possible aerogel upgrade
is under investigation.

STOPGAP detector. This proposed device takes advantage of the development of
fast CMOS sensors. Two ideas are discussed, one being to fill the gaps in the
TOP detector with a $\approx 1$ cm2 granularity to improve the kaon detection
efficiency by about 10% by covering the full solid angle. In the second
proposal, two full timing layers are added at lower radii ($\approx 250$ mm,
$\approx 450$ mm) to provide PID for low-momentum particles in the context of
a larger VXD detector; such a detector could also provide trigger information.
This interesting concept for longer-term upgrades requires more R&D effort.

###  3.4 Electromagnetic calorimetry

The electromagnetic calorimeter performs very well despite of the larger
background levels compared to the Belle experiment. Only possible long-term
upgrades are therefore discussed. In one of the scenarios, the present
scintillator crystals - CsI(Tl) - would be replaced by a considerably faster
pure CsI scintillator, reducing the pile-up. A wavelength-shifting (WLS)
window would shift the wavelength of the scintillation light to match the PiN
diode sensitivity. In the second scenario, a pre-shower detector would be
installed to reduce the background and pile-up and to determine photon
direction and timing. The third option foresees a replacement of the present
scintillation light sensors (PiN diodes) with APDs; this would improve the
resolution and reduce the noise. We note that all three options are complex
and expensive and are thus only considered in long-term planning.

###  3.5 Muon and $K_{L}$ detector: KLM

In the muon detection system, a replacement of the RPC detectors with
scintillator bars, WLS fibers, and SiPMs as light sensors is discussed for the
entire system in a similar way as already carried out for the endcaps and the
first two layers in the barrel part of the spectrometer. By this, we would
increase the rate capability of the system. This would be accompanied by a
read-out electronics upgrade, with a more compact read-out and a possible
data-push architecture. Possible use as a TOF detector is also discussed; a
time resolution of around 100 ps would improve the $K_{L}$ identification.

###  3.6 Trigger

The trigger system upgrade foresees installing a more powerful UT4 board for
new CDC Front End electronics. Such an upgrade would allow for more bandwidth
and the use of all CDC TDC and ADC information and would make many trigger
improvements possible. Detailed technical documents on this upgrade are in
preparation.

###  3.7 Polarized electron beam

An exciting possibility of further expanding the experiment’s physics reach
would be using a polarized beam [4]. The plan is to reach a 70% polarization
with an 80% polarized source. New hardware for the polarization upgrade would
include a low-emittance polarized source, spin rotators, and a Compton
polarimeter to monitor the longitudinal polarization. Beam polarization
options at SuperKEKB are under active study; there are some indications that
such a modification may be possible on a more rapid timescale.

##  4 Summary

Belle II and SuperKEKB have started a successful physics run. Machine
improvements are being studied and implemented to reach the target luminosity.
Detector upgrade ideas are being explored, and R&D is in progress to achieve
better robustness against background and radiation damage, improve the physics
performance, and be ready for a possible interaction region redesign.

The Belle II upgrade organization is in place. The Upgrade Working Group and
Upgrade Advisory Committee have been set up to help establish priorities and
direct the effort. A Belle II Upgrades White paper has been submitted to the
US Snowmass process.[7] The transition to a construction project is needed
soon. While the SuperKEKB International Task Force is finalizing its
recommendations, the preparation of an Upgrade Conceptual Design Report has
started with the aim to be ready end of 2023.

For a longer-term perspective of the Belle II experiment, it is important to
start exploring the options. It is worth noting that there is considerable
physics potential at even higher luminosities and with an even better
detector.

## References

  * [1] T. Abe et al., ”Belle II Technical Design Report,” KEK Report 2010-1 (2010)
  * [2] I. Adachi et al., ”Detectors for extreme luminosity: Belle II,” Nucl. Instrum. Meth. A 907 (2018) 46
  * [3] Y. Ohnishi, T. Abe, T. Adachi, K. Akai, Y. Arimoto, K. Ebihara, K. Egawa, J. Flanagan, H. Fukuma and Y. Funakoshi, et al. “Accelerator design at SuperKEKB,” PTEP 2013 (2013) 03A011
  * [4] E. Kou et al., ”The Belle II Physics Book,” PTEP 2019 (2019) 123C01
  * [5] Z. J. Liptak, A. Paladino, L. Šantelj, J. Schueler, S. Stefkova, H. Tanigawa, N. Tsuzuki, A. Aloisio, P. Ahlburg and P. Bambade, et al. “Measurements of beam backgrounds in SuperKEKB Phase 2,” Nucl. Instrum. Meth. A 1040 (2022) 167168
  * [6] I. Adachi et al. [Belle-II], “Search for an Invisibly Decaying $Z^{\prime}$ Boson at Belle II in $e^{+}e^{-}\to\mu^{+}\mu^{-}(e^{\pm}\mu^{\mp})$ Plus Missing Energy Final States,” Phys. Rev. Lett. 124 (2020) 141801
  * [7] F. Forti [Belle-II], “Snowmass Whitepaper: The Belle II Detector Upgrade Program,” [arXiv:2203.11349 [hep-ex]].


#  A Relationship for LYM Inequalities between
Boolean Lattices and Linear Lattices with Applications

Jiuqiang Liua,b,, Guihai Yua,∗
a College of Big Data Statistics, Guizhou University of Finance and Economics
Guiyang, Guizhou, 550025, China
b Department of Mathematics, Eastern Michigan University
Ypsilanti, MI 48197, USA
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>The corresponding author

###### Abstract

Sperner theory is one of the most important branches in extremal set theory.
It has many applications in the field of operation research, computer science,
hypergraph theory and so on. The LYM property has become an important tool for
studying Sperner property. In this paper, we provide a general relationship
for LYM inequalities between Boolean lattices and linear lattices. As
applications, we use this relationship to derive generalizations of some well-
known theorems on maximum sizes of families containing no copy of certain
poset or certain configuration from Boolean lattices to linear lattices,
including generalizations of the well-known Kleitman theorem on families
containing no $s$ pairwise disjoint members (a non-uniform variant of the
famous Erdős matching conjecture) and Johnston-Lu-Milans theorem and Polymath
theorem on families containing no $d$-dimensional Boolean algebras.

Key words: Extremal combinatorics; Erdős-Ko-Rado theorem; Erdős matching
conjecture; LYM inequality; Sperner theory
AMS Classifications: 05D05.

##  1 Introduction

Throughout this paper, we denote $[n]=\\{1,2,\dots,n\\}$. A Boolean lattice
$\mathcal{B}_{n}$ is the set $2^{[n]}$ of all subsets of $[n]$ with the
ordering being the containment relation and a linear lattice
$\mathcal{L}_{n}(q)$ is the set of all subspaces of the $n$-dimensional vector
space $\mathbb{F}_{q}^{n}$ over the finite field $\mathbb{F}_{q}$ with the
ordering being the relation of inclusion of subspaces. Denote by
${{[n]}\choose{k}}$ for the set of all $k$-subsets of $[n]$ and
$\genfrac{[}{]}{0.0pt}{}{[n]}{i}$ denotes the set of all $i$-dimensional
subspaces of $\mathbb{F}_{q}^{n}$.

Given posets $P$ and $P^{\prime}$, we say that $P$ is weakly contained in
$P^{\prime}$ if there exists an injective map $\psi:P\longrightarrow
P^{\prime}$ such that for every $u,v\in P$, $\psi(u)\leq_{P^{\prime}}\psi(v)$
if $u\leq_{P}v$; and $P^{\prime}$ strongly contains $P$ if for all $u,v\in P$,
$\psi(u)\leq_{P^{\prime}}\psi(v)$ if and only if $u\leq_{P}v$.

Given a poset $P$, $ex(n,P)$ is defined by

$ex(n,P)=\max\\{|\mathcal{F}|\mid\mathcal{F}\subseteq 2^{[n]}\mbox{ and
}\mathcal{F}\mbox{ does not contain weakly }P\\}$

and $en^{*}(n,P)$ is defined by

$ex^{*}(n,P)=\max\\{|\mathcal{F}|\mid\mathcal{F}\subseteq 2^{[n]}\mbox{ and
}\mathcal{F}\mbox{ does not contain strongly }P.\\}$

There has been a considerable amount of research devoted to determining the
asymptotic behaviour of $ex(n,P)$ and $ex^{*}(n,P)$. The first result of this
kind is the following well-known Sperner’s Theorem. Recall that a family of
subsets of $[n]$ is called a Sperner family (or antichain) if there are no two
different members of the family such that one of them contains the other.

###### Theorem 1.1

(Sperner Theorem). Suppose that $\mathcal{A}$ is a Sperner family of subsets
of $[n]$. Then

$|\mathcal{A}|\leq{{n}\choose{\lfloor\frac{n}{2}\rfloor}}.$

Clearly,, Sperner’s result is equivalent to say that
$ex(n,P_{2})={{n}\choose{\lfloor\frac{n}{2}\rfloor}}$.

We say a family $\mathcal{F}$ of subsets of $[n]$ is $k$-$Sperner$ if all
chains in $\mathcal{F}$ have length at most $k$. Define $\sum(n,k)$ to be the
sum of the $k$ largest binomial coefficients of order $n$, i.e.,
$\sum(n,k)=\sum_{i=1}^{k}{{n}\choose{\lfloor\frac{n-k}{2}\rfloor+i}}$. Let
$\sum^{*}(n,k)$ be the collection of families consisting of the corresponding
full levels, i.e., if $n+k$ is odd, then $\sum^{*}(n,k)$ contains one family
$\cup_{i=1}^{k}{{[n]}\choose{\lfloor\frac{n-k}{2}\rfloor+i}}$; if $n+k$ is
even, then $\sum^{*}(n,k)$ contains two families of the same size
$\cup_{i=0}^{k-1}{{[n]}\choose{\frac{n-k}{2}+i}}$ and
$\cup_{i=1}^{k}{{[n]}\choose{\frac{n-k}{2}+i}}$. The following theorem by
Erdős [7] generalizes the classical Sperner theorem.

###### Theorem 1.2

(Erdős, [7]). Suppose that $\mathcal{A}$ is a $k$-Sperner family of subsets of
$[n]$. Then

$|\mathcal{A}|\leq\sum(n,k).$

Moreover, if $|\mathcal{A}|=\sum(n,k)$, then $\mathcal{A}\in\sum^{*}(n,k)$.

Denote

$\genfrac{[}{]}{0.0pt}{}{n}{k}=\prod_{0\leq i\leq
k-1}\frac{q^{n-i}-1}{q^{k-i}-1}.$ (1)

It is well known that the number of all $k$-dimensional subspaces of
$\mathbb{F}_{q}^{n}$ is equal to $\genfrac{[}{]}{0.0pt}{}{n}{k}$.

Note that there has been a considerable amount of research on maximum sizes of
families of subsets of $[n]$ forbidden certain structures and many of the
results appeared in the literature have their $q$-analogues (generalizations
to linear lattices) which are usually derived using different and more
complicated methods, see for example [1], [4], [9], [13], [15], [19], [23],
[34], and [35].

We say that a family $\mathcal{V}$ of subspaces is Sperner (or antichain) if
no subspace is contained in another subspace in $\mathcal{V}$, and
$\mathcal{V}$ is $k$-Sperner if all chains in $\mathcal{V}$ have length at
most $k$. The following well-known Sperner theorem for families of subspaces
of $\mathbb{F}_{q}^{n}$ can be found in [6], which is a vector space analogue
(or $q$-analogue) of the classical Sperner theorem.

###### Theorem 1.3

(q-Analogue Sperner Theorem). Assume that $\mathcal{V}$ is a Sperner family of
subspaces of $\mathbb{F}_{q}^{n}$. Then

$|\mathcal{V}|\leq\genfrac{[}{]}{0.0pt}{}{n}{\lfloor\frac{n}{2}\rfloor}.$

We denote $\sum[n,k]$ to be the sum of the $k$ largest $q$-binomial
coefficients of order $n$, i.e.,
$\sum[n,k]=\sum_{i=1}^{k}\left[n\atop\lfloor\frac{n-k}{2}\rfloor+i\right]$.
Let $\sum^{*}[n,k]$ be the collection of families consisting of the
corresponding full levels, i.e., if $n+k$ is odd, then $\sum^{*}[n,k]$
contains one family
$\cup_{i=1}^{k}\left[[n]\atop\lfloor\frac{n-k}{2}\rfloor+i\right]$; if $n+k$
is even, then $\sum^{*}[n,k]$ contains two families of the same size
$\cup_{i=0}^{k-1}\left[[n]\atop\frac{n-k}{2}+i\right]$ and
$\cup_{i=1}^{k}\left[[n]\atop\frac{n-k}{2}+i\right]$.

The following $q$-analogue of Erdős’ theorem (Theorem 1.2) is implied by
Theorem 2 in [36], which generalizes Theorem 1.3 to $k$-Sperner families.

###### Theorem 1.4

(Samotij, [36]). Assume that $\mathcal{V}$ is a $k$-Sperner family of
subspaces of $\mathbb{F}_{q}^{n}$. Then

$|\mathcal{V}|\leq\sum[n,k].$

Moreover, if $|\mathcal{V}|=\sum[n,k]$, then $\mathcal{V}\in\sum^{*}[n,k]$.

Given $\mathcal{F}\subseteq 2^{[n]}$, the Lubell function of $\mathcal{F}$,
also called weight and denoted by $l(\mathcal{F})$, is the quantity

$l(\mathcal{F})=\sum_{F\in\mathcal{F}}\frac{1}{{{n}\choose{|F|}}}=\sum_{i=0}^{n}\frac{|\mathcal{F}_{i}|}{{{n}\choose{i}}},$
(2)

where, $\mathcal{F}_{i}=\\{F\in\mathcal{F}\mid|F|=i\\}$.

One of the advantages of using the Lubell function is its convenient
probabilistic interpretation: Suppose that
$\mathcal{C}=\\{\emptyset,\\{i_{1}\\},\\{i_{1},i_{2}\\},\dots,[n]\\}$ is full-
chain in $2^{[n]}$ chosen uniformly at random. Let $X$ be the random variable
$X=|\mathcal{C}\cap\mathcal{F}|$. Then one has the expected value
$E(X)=l(\mathcal{F})$.

The following well-known LYM inequality was proved independently by Bollobás
[3], Lubell [29], Meshalkin [32] and Yamamoto [38].

###### Theorem 1.5

(LYM Inequality, Bollobás [3], Lubell [29], Meshalkin [32] and Yamamoto [38]).
If $\mathcal{F}\subseteq 2^{[n]}$ is an antichain, then

$l(\mathcal{F})=\sum_{F\in\mathcal{F}}\frac{1}{{{n}\choose{|F|}}}\leq 1.$

This LYM inequality naturally implies Sperner’s Theorem (Theorem 1.1) and is a
cornerstone of the field of extremal set theory, and the weight or Lubell
function $l(\mathcal{F})$ is a powerful tool for studying both $ex(n,P)$ and
$ex^{*}(n,P)$ and has been used in various papers, see for example [20], [21],
[24], and [27]. The LYM inequality has been strengthened by Bey [2] and Erdős
et al. [11] and has a continuous analogue by Klain and Rota [25].

In general, given a ranked poset $\mathcal{P}$, in which the set of elements
of rank $r$ is denoted by $\mathcal{P}_{r}$, $\mathcal{P}$ has the Sperner
property if the largest antichain has size equal to the size of the largest
rank level, and $\mathcal{P}$ satisfies the LYM property (or LYM inequality)
if for every antichain $\mathcal{A}$ of $\mathcal{P}$,

$\sum_{i}\frac{|\mathcal{A}\cap\mathcal{P}_{i}|}{|\mathcal{P}_{i}|}\leq 1.$
(3)

Clearly, for $\mathcal{P}=\mathcal{B}_{n}$, the inequality (3) is the same as
the inequality in Theorem 1.5 with $\mathcal{P}_{i}={{[n]}\choose{i}}$.

For $k$-Sperner families, one can show the next fact.

###### Theorem 1.6

If $\mathcal{F}\subseteq 2^{[n]}$ contains no $P_{k+1}$, then

$l(\mathcal{F})\leq k.$

This inequality implies Erdős $k$-Sperner theorem (Theorem 1.2). In 2018,
M$\grave{e}$roueh [31] proved the following general result which was
conjectured to be true by Lu and Milans [28].

###### Theorem 1.7

(M$\grave{e}$roueh, [31]). For every poset $P$, there exists $c(P)$ such that
if $\mathcal{F}\subseteq 2^{[n]}$ for some $n\in\mathbb{N}$ and $\mathcal{F}$
does not contain $P$ as an induced subposet, then $l(\mathcal{F})\leq c(P)$.

Clearly, a family $\mathcal{F}\subseteq 2^{[n]}$ does not contain $P$ as a
subposet implies it does not contain $P$ as an induced subposet. Theorem 1.7
has the next immediate consequence.

###### Theorem 1.8

For every poset $P$, there exists $c(P)$ such that if $\mathcal{F}\subseteq
2^{[n]}$ for some $n\in\mathbb{N}$ and $\mathcal{F}$ does not contain $P$ as a
subposet, then $l(\mathcal{F})\leq c(P)$.

Our first main result is the following general relationship for LYM
inequalities between Boolean lattices and linear lattices (many results about
maximum sizes of families of subspaces of vector space $\mathbb{F}_{q}^{n}$
can be derived from the corresponding theorems for maximum sizes of families
of subsets of $[n]$ through this relationship). For a family $\mathcal{V}$ of
subspaces of $\mathbb{F}_{q}^{n}$, we denote

$l_{q}(\mathcal{V})=\sum_{V\in\mathcal{V}}\frac{1}{\genfrac{[}{]}{0.0pt}{}{n}{|V|}}=\sum_{i=0}^{n}\frac{|\mathcal{V}_{i}|}{\genfrac{[}{]}{0.0pt}{}{n}{i}},$
(4)

where $\mathcal{V}_{i}=\\{V\in\mathcal{V}\mid dim(V)=i\\}$.

###### Theorem 1.9

Let $P$ be a given poset or configuration. Suppose that there exists $c(P)$
such that if $\mathcal{F}\subseteq 2^{[n]}$ and $\mathcal{F}$ does not contain
$P$, then $l(\mathcal{F})\leq c(P)$. Then every family $\mathcal{V}$ of
subspaces of $\mathbb{F}_{q}^{n}$ which contains no copy of $P$ satisfies
$l_{q}(\mathcal{V})=\sum_{V\in\mathcal{V}}\frac{1}{\genfrac{[}{]}{0.0pt}{}{n}{|V|}}\leq
c(P)$.

Clearly, Theorems 1.8 and 1.9 imply the next consequence which is a
generalization of Theorem 1.8 to linear lattices.

###### Theorem 1.10

For every poset $P$, there exists $c(P)$ such that every family $\mathcal{V}$
of subspaces of $\mathbb{F}_{q}^{n}$ which contains no copy of $P$ satisfies
$l_{q}(\mathcal{V})=\sum_{V\in\mathcal{V}}\frac{1}{\genfrac{[}{]}{0.0pt}{}{n}{|V|}}\leq
c(P)$.

Define the weighted Lubell functions $l^{*}(\mathcal{F})$ and
$l^{*}_{q}(\mathcal{V})$ by

$l^{*}(\mathcal{F},\beta)=\sum_{i=0}^{n}\frac{\beta_{i}|\mathcal{F}_{i}|}{{{n}\choose{i}}},$
$l^{*}_{q}(\mathcal{V},\beta)=\sum_{i=0}^{n}\frac{\beta_{i}|\mathcal{V}_{i}|}{\genfrac{[}{]}{0.0pt}{}{n}{i}},$

where $\mathcal{F}_{i}=\mathcal{F}\cap{{[n]}\choose{i}}$ and
$\mathcal{V}_{i}=\mathcal{V}\cap\genfrac{[}{]}{0.0pt}{}{[n]}{i}$. We have the
next weighted version of Theorem 1.9.

###### Theorem 1.11

Let $P$ be a given poset or configuration. Suppose that there exists $c(P)$
such that if $\mathcal{F}\subseteq 2^{[n]}$ and $\mathcal{F}$ does not contain
$P$, then $l^{*}(\mathcal{F},\beta)\leq c(P)$. Then every family $\mathcal{V}$
of subspaces of $\mathbb{F}_{q}^{n}$ which contains no copy of $P$ satisfies
$l^{*}_{q}(\mathcal{V},\beta)\leq c(P)$.

As applications of Theorem 1.9, in sections 3 - 5, we derive generalizations
of some well-known theorems on maximum sizes of families containing no copy of
certain poset or certain configuration from Boolean lattices to linear
lattices, including generalizations of Grósz-Methuku-Tompkins theorem on
diamond-free families, generalizations of Johnston-Lu-Milans theorem and
Polymath theorem on families containing no $d$-dimensional Boolean algebras
which is used by Polymath in a new proof of Furstenberg and Katznelson’s
density Hales-Jewett theorem (a generalization of the well-known Szemerédi
theorem), and generalizations of the following well-known Kleitman theorem on
families containing no $s$ pairwise disjoint members (a non-uniform variant of
the well-known Erdős Matching Conjecture [8]) and Frankl-Kupavskii theorem on
cross-dependent families [14] as shown in Theorems 1.13 and 1.14 below.

###### Theorem 1.12

(Kleitman, [26]). Let $n\geq s\geq 3$. Suppose that $\mathcal{K}\subseteq
2^{[n]}$ contains no $s$ pairwise disjoint members. Then the following (i) and
(ii) hold:

(i) If $n=sk-1$, then

$|\mathcal{K}|\leq\sum_{i\geq k}{{n}\choose{i}};$

(ii) If $n=sk+r$ with $0\leq r\leq s-2$, then

$|\mathcal{K}|\leq\sum_{i>k}{{n}\choose{i}}+\frac{s-r-1}{s}{{n}\choose{k}}.$

We say two subspaces $V_{1},V_{2}$ in the vector space $\mathbb{F}_{q}^{n}$
are disjoint if $dim(V_{1}\cap V_{2})=0$. The families
$\mathcal{V}_{1},\mathcal{V}_{2},\dots,\mathcal{V}_{s}$ of subspaces of
$\mathbb{F}_{q}^{n}$ are called cross-dependent if there is no choice of
$V_{1}\in\mathcal{V}_{1},V_{2}\in\mathcal{V}_{2},\dots,V_{s}\in\mathcal{V}_{s}$
such that $V_{1},V_{2},\dots,V_{s}$ are pairwise disjoint.

###### Theorem 1.13

Let $n\geq s\geq 3$. Suppose that $\mathcal{V}\subseteq\mathcal{L}_{n}(q)$
contains no $s$ pairwise disjoint members. Then the following (i) and (ii)
hold:

(i) If $n=sk-1$, then

$|\mathcal{V}|\leq\sum_{i\geq k}\genfrac{[}{]}{0.0pt}{}{n}{i};$ (5)

(ii) If $n=sk+r$ with $0\leq r\leq s-2$, then

$|\mathcal{V}|\leq\sum_{i>k}\genfrac{[}{]}{0.0pt}{}{n}{i}+\frac{s-r-1}{s}\genfrac{[}{]}{0.0pt}{}{n}{k}.$
(6)

Moreover, the bound in (5) is sharp for the case $n=sk-1$.

###### Theorem 1.14

Suppose that $n=sl+r$ where $0\leq r<s$ and $s\geq 3$. Let
$\mathcal{V}_{1},\mathcal{V}_{2},\dots,\mathcal{V}_{s}\subseteq\mathcal{L}_{n}(q)$
be cross-dependent. Then

$\sum_{1\leq i\leq s}|\mathcal{V}_{i}|\leq s\sum_{l<j\leq
n}\genfrac{[}{]}{0.0pt}{}{n}{j}+(s-r-1)\genfrac{[}{]}{0.0pt}{}{n}{l}.$ (7)

Moreover, the bound in (7) is sharp.

Clearly, Theorem 1.14 gives rise to Theorem 1.13 by taking
$\mathcal{V}_{1}=\mathcal{V}_{2}=\cdots=\mathcal{V}_{s}=\mathcal{V}$. We will
apply Theorem 1.9 to derive Theorems 1.13 and 1.14 in section 4.

##  2 Proofs of Theorems 1.9 and 1.11

First, we introduce some notions and the following covering lemma by Gerbner
[17]. Let $S$ be a set and $S=S_{0}\cup S_{1}\cup\cdots\cup S_{n}$ be a
partition (In the discussions here, $S$ is either the set $2^{[n]}$ of all
subsets of $[n]$ or the set of all subspaces of $\mathbb{F}_{q}^{n}$, and
$S_{i}$ will be level $i$, i.e., $S_{i}={{[n]}\choose{i}}$ or the set of all
$i$-dimensional subspaces of $\mathbb{F}_{q}^{n}$, respectively). Given a
vector $t=(t_{0},t_{1},\dots,t_{n})$, we say a family $\Gamma$ of subsets of
$S$ is a $t$-covering family of $S$ if for each $0\leq i\leq n$, each member
in $S_{i}$ is contained in exactly $t_{i}$ sets in the family $\Gamma$.

Given a family $\mathcal{F}\subseteq S$, let $f_{i}=|\mathcal{F}\cap S_{i}|$
and $(f_{0},f_{1},\dots,f_{n})$ is called the profile vector of $\mathcal{F}$.
For a weight $\overline{w}=(w_{0},w_{1},\dots,w_{n})$ and a family
$\mathcal{F}\subseteq S$, let
$\overline{w}(\mathcal{F})=\sum_{i=0}^{n}w_{i}|\mathcal{F}\cap S_{i}|$. Denote
$\overline{w/t}=(w_{0}/t_{0},w_{1}/t_{1},\dots,w_{n}/t_{n})$. The following
lemma is Lemma 2.1 in [17].

We say that a property $P$ of families is hereditary if for any family
$\mathcal{F}$ with property $P$, every subfamily of $\mathcal{F}$ has property
$P$. Clearly, forbidding a poset or a configuration is a hereditary property.

###### Lemma 2.1

(Gerbner [17]). Let $P$ be a hereditary property of subsets (families) of $S$
and $\Gamma$ be a $t$-covering family of $S$. Assume that there exists a real
number $x$ such that for every $G\in\Gamma$, every subset $G^{\prime}$ of $G$
with property $P$ has $\overline{w/t}(G^{\prime})\leq x$. Then
$\overline{w}(F)\leq|\Gamma|x$ for every $F\subseteq S$ with property $P$.

Let $S=\mathcal{L}_{n}(q)$ be the set of all subspaces of
$\mathbb{F}_{q}^{n}$. Denote $S=S_{0}\cup S_{1}\cup\cdots\cup S_{n}$ for a
partition of $S$ such that $S_{i}$ is the set of all $i$-dimensional subspaces
of $\mathbb{F}_{q}^{n}$. We construct a $t$-covering family $\Gamma$ of $S$
such that every $\mathcal{G}\in\Gamma$ is a subfamily in $\mathcal{L}_{n}(q)$
isomorphic to Boolean lattice $\mathcal{B}_{n}$ as follows: Choose an
arbitrary basis $B=\\{v_{1},\dots,v_{n}\\}$ of $\mathbb{F}_{q}^{n}$ and let
$\mathcal{G}_{B}=\\{span(U)\mid U\subseteq B\\}$, i.e., the family of all
subspaces that are generated by subsets of the vectors in $B$. Obviously the
function that maps $H\subseteq[n]$ to the subspace $span\\{v_{x}\mid x\in
H\\}$ keeps inclusion and intersection properties. Let $\Gamma$ be the
collection of the families $\mathcal{G}_{B}$ over all bases $B$, i.e.,
$\Gamma=\\{\mathcal{G}_{B}\mid B\mbox{ is a basis of }\mathbb{F}_{q}^{n}\\}$.
Denote

$\alpha(q,n)=\frac{(q^{n}-1)(q^{n}-q)(q^{n}-q^{2})\cdots(q^{n}-q^{n-1})}{n!}.$

Let $t=(t_{0},t_{1},\dots,t_{n})$ be such that for $0\leq i\leq n$,

$t_{i}=\frac{(q^{i}-1)(q^{i}-q)\cdots(q^{i}-q^{i-1})(q^{n}-q^{i})\cdots(q^{n}-q^{n-1})}{i!(n-i)!}.$
(8)

The next lemma follows from an easy counting.

###### Lemma 2.2

Let $S=\mathcal{L}_{n}(q)$ be the set of all subspaces of $\mathbb{F}_{q}^{n}$
and $\Gamma=\\{\mathcal{G}_{B}\mid B\mbox{ is a basis of
}\mathbb{F}_{q}^{n}\\}$. Then $|\Gamma|=\alpha(q,n)$ and $\Gamma$ is a
$t$-covering of $S$ with $t=(t_{0},t_{1},\dots,t_{n})$ given in (8).

Proof. To show $|\Gamma|=\alpha(q,n)$, it suffices to show that
$\mathbb{F}_{q}^{n}$ has $\alpha(q,n)$ different bases. To obtain a basis
$B=\\{v_{1},v_{2},\dots,v_{n}\\}$ for $\mathbb{F}_{q}^{n}$, one can choose $n$
nonzero linearly independent vectors in the order $v_{1},v_{2},\dots,v_{n}$ so
that $v_{j}\not\in span\\{v_{1},v_{2},\dots,v_{j-1}\\}$ as follows: There are
$q^{n}-1$ ways to choose $v_{1}$, and then there are $q^{n}-q$ ways to choose
$v_{2}$, $q^{n}-q^{2}$ ways to choose $v_{3}$, …, $q^{n}-q^{n-1}$ ways to
choose $v_{n}$. Thus, there are
$(q^{n}-1)(q^{n}-q)(q^{n}-q^{2})\cdots(q^{n}-q^{n-1})$ combined ways to form
$B=\\{v_{1},v_{2},\dots,v_{n}\\}$ (in order). Clearly, there are $n!$ ways
(permutations on $B$) to form the same $B$ (without order). It follows that
there are

$\alpha(q,n)=\frac{(q^{n}-1)(q^{n}-q)(q^{n}-q^{2})\cdots(q^{n}-q^{n-1})}{n!}$

different bases in $\mathbb{F}_{q}^{n}$.

For any $i$-dimensional subspace $V$ in $\mathbb{F}_{q}^{n}$, similar to the
argument above, there are

$\frac{(q^{i}-1)(q^{i}-q)\cdots(q^{i}-q^{i-1})}{i!}$

different bases in $V$. For each basis $D$ in $V$, one can extend $D$ to a
basis of $\mathbb{F}_{q}^{n}$ in

$\frac{(q^{n}-q^{i})(q^{n}-q^{i+1})\cdots(q^{n}-q^{n-1})}{(n-i)!}$

different ways. Therefore, $V$ is contained in

$t_{i}=\frac{(q^{i}-1)(q^{i}-q)\cdots(q^{i}-q^{i-1})(q^{n}-q^{i})\cdots(q^{n}-q^{n-1})}{i!(n-i)!}$

different $\mathcal{G}_{B}\in\Gamma$. It follows that $\Gamma$ is a
$t$-covering of $S=\mathcal{L}_{n}(q)$. $\hfill\Box$

We now provide a proof for Theorem 1.9.

Proof of Theorem 1.9. Assume that $\mathcal{V}$ is a family in
$\mathcal{L}_{n}(q)$ which contains no copy of poset $P$. Let
$\Gamma=\\{\mathcal{G}_{B}\mid B\mbox{ is a basis of }\mathbb{F}_{q}^{n}\\}$.
By Lemma 2.2, $\Gamma$ is a $t$-covering of $S=\mathcal{L}_{n}(q)$. Let
$w=(w_{0},w_{1},\dots,w_{n})$ with $w_{i}=\frac{t_{i}}{{{n}\choose{i}}}$ for
each $0\leq i\leq n$. Then

$\overline{w/t}=\bigg{(}\frac{1}{{{n}\choose{0}}},\frac{1}{{{n}\choose{1}}},\dots,\frac{1}{{{n}\choose{n}}}\bigg{)}.$

By the construction, every $\mathcal{G}\in\Gamma$ is isomorphic to Boolean
lattice $\mathcal{B}_{n}$. Let $\mathcal{G}^{\prime}\subseteq\mathcal{G}$ be
any subfamily without containing a copy of $P$ and let

$|\mathcal{G}^{\prime}|=a_{0}{{n}\choose{0}}+a_{1}{{n}\choose{1}}+\cdots+a_{n}{{n}\choose{n}},$

where $a_{i}{{n}\choose{i}}=|\mathcal{G}^{\prime}_{i}|$ and
$\mathcal{G}^{\prime}_{i}$ is the set of all $i$-dimensional subspaces in
$\mathcal{G}^{\prime}$. Then

$l(\mathcal{G}^{\prime})=a_{0}+a_{1}+\cdots+a_{n}.$

By the assumption,

$\overline{w/t}(\mathcal{G}^{\prime})=a_{0}{{n}\choose{0}}\cdot\frac{1}{{{n}\choose{0}}}+a_{1}{{n}\choose{1}}\cdot\frac{1}{{{n}\choose{1}}}+\cdots+a_{n}{{n}\choose{n}}\cdot\frac{1}{{{n}\choose{n}}}=a_{0}+a_{1}+\cdots+a_{n}=l(\mathcal{G}^{\prime})\leq
c(P).$

By Lemma 2.1, we have

$\overline{w}(\mathcal{V})\leq|\Gamma|\cdot c(P).$ (9)

Let

$|\mathcal{V}|=b_{0}\genfrac{[}{]}{0.0pt}{}{n}{0}+b_{1}\genfrac{[}{]}{0.0pt}{}{n}{1}+\cdots+b_{n}\genfrac{[}{]}{0.0pt}{}{n}{n}$

with $b_{i}\genfrac{[}{]}{0.0pt}{}{n}{i}$ being the number of $i$-dimensional
subspaces in $\mathcal{V}$. Then

$l_{q}(\mathcal{V})=b_{0}+b_{1}+\cdots+b_{n}.$

It follows that

$\overline{w}(\mathcal{V})=\sum_{i=0}^{n}b_{i}w_{i}\genfrac{[}{]}{0.0pt}{}{n}{i}\hskip
278.837pt$
$=\sum_{i=0}^{n}\frac{b_{i}}{{{n}\choose{i}}}\frac{(q^{i}-1)(q^{i}-q)\cdots(q^{i}-q^{i-1})(q^{n}-q^{i})\cdots(q^{n}-q^{n-1})}{i!(n-i)!}\genfrac{[}{]}{0.0pt}{}{n}{i}$
$=\sum_{i=0}^{n}\frac{b_{i}}{{{n}\choose{i}}}\frac{(q^{i}-1)\cdots(q^{i}-q^{i-1})(q^{n}-q^{i})\cdots(q^{n}-q^{n-1})}{i!(n-i)!}\hskip
54.06023pt$ $\hskip
159.33542pt\cdot\frac{(q^{n}-1)(q^{n}-q)\cdots(q^{n}-q^{n-1})}{(q^{i}-1)\cdots(q^{i}-q^{i-1})(q^{n}-q^{i})\cdots(q^{n}-q^{n-1})}$
$=\sum_{i=0}^{n}\frac{b_{i}}{{{n}\choose{i}}}\frac{\alpha(n,q)n!}{i!(n-i)!}=\sum_{i=0}^{h}\frac{b_{i}}{{{n}\choose{i}}}{{n}\choose{i}}|\Gamma|\hskip
139.4185pt$ $=\sum_{i=0}^{n}b_{i}|\Gamma|=l_{q}(\mathcal{V})|\Gamma|.\hskip
210.55038pt$

Combining with (9), we obtain

$l_{q}(\mathcal{V})|\Gamma|\leq|\Gamma|\cdot c(P),\mbox{ that is,
}l_{q}(\mathcal{V})\leq c(P).$

$\hfill\Box$

Proof of Theorem 1.11. The proof is obtained by modifying the proof of Theorem
1.9 above. Indeed, let $w=(w_{0},w_{1},\dots,w_{n})$ with
$w_{i}=\frac{\beta_{i}t_{i}}{{{n}\choose{i}}}$. We have

$\overline{w/t}=\bigg{(}\frac{\beta_{0}}{{{n}\choose{0}}},\frac{\beta_{1}}{{{n}\choose{1}}},\dots,\frac{\beta_{n}}{{{n}\choose{n}}}\bigg{)},$
$l^{*}(\mathcal{G}^{\prime},\beta)=\sum_{i=0}^{n}\frac{\beta_{i}|\mathcal{G}^{\prime}_{i}|}{{{n}\choose{i}}}=a_{0}\beta_{0}+a_{1}\beta_{1}+\cdots+a_{n}\beta_{n},$
$\overline{w/t}(\mathcal{G}^{\prime})=a_{0}{{n}\choose{0}}\cdot\frac{\beta_{0}}{{{n}\choose{0}}}+a_{1}{{n}\choose{1}}\cdot\frac{\beta_{1}}{{{n}\choose{1}}}+\cdots+a_{n}{{n}\choose{n}}\cdot\frac{\beta_{n}}{{{n}\choose{n}}}\hskip
22.76219pt$
$=a_{0}\beta_{0}+a_{1}\beta_{1}+\cdots+a_{n}\beta_{n}=l^{*}(\mathcal{G}^{\prime},\beta)\leq
c(P).$

By Lemma 2.1, we have

$\overline{w}(\mathcal{V})\leq|\Gamma|\cdot c(P).$ (10)

On the other hand, we have

$\overline{w}(\mathcal{V})=\sum_{i=0}^{n}b_{i}w_{i}\genfrac{[}{]}{0.0pt}{}{n}{i}\hskip
284.52756pt$
$=\sum_{i=0}^{n}\frac{b_{i}\beta_{i}}{{{n}\choose{i}}}\frac{(q^{i}-1)(q^{i}-q)\cdots(q^{i}-q^{i-1})(q^{n}-q^{i})\cdots(q^{n}-q^{n-1})}{i!(n-i)!}\genfrac{[}{]}{0.0pt}{}{n}{i}$
$=\sum_{i=0}^{n}b_{i}\beta_{i}|\Gamma|=l^{*}_{q}(\mathcal{V},\beta)|\Gamma|.\hskip
193.47873pt$

Together with (10), it follows that

$l^{*}_{q}(\mathcal{V},\beta)\leq c(P).$

$\hfill\Box$

##  3 Sperner families and Diamond-free families

In this section, we apply Theorems 1.9 and 1.11 to derive generalizations of
results on Sperner families and diamond-free families of subsets of $[n]$.

###  3.1 Sperner and $k$-Sperner families

Clearly, Theorems 1.5 and 1.9 give the next generalization of Theorem 1.5
which implies Theorem 1.3.

###### Theorem 3.1

(q-Analogue LYM inequality). Assume that $\mathcal{V}$ is a Sperner family of
subspaces of $\mathbb{F}_{q}^{n}$. Then

$l_{q}(\mathcal{V})=\sum_{V\in\mathcal{V}}\frac{1}{\genfrac{[}{]}{0.0pt}{}{n}{|V|}}\leq
1.$

Also, Theorems 1.6 and 1.9 give the following generalization of Theorem 1.6
which implies Theorem 1.4.

###### Theorem 3.2

Assume that $\mathcal{V}$ is a $k$-Sperner family of subspaces of
$\mathbb{F}_{q}^{n}$. Then

$l_{q}(\mathcal{V})=\sum_{V\in\mathcal{V}}\frac{1}{\genfrac{[}{]}{0.0pt}{}{n}{|V|}}\leq
k.$

###  3.2 Diamond-free families

The diamond poset, denoted by $\mathcal{Q}_{2}$, is defined on four elements
$x,y,z,w$ with the relations $x<y,z$ and $y,z<w$. Recall that $ex(n,P)$ is the
maximum size of a family of subsets of $[n]$ not containing $P$,
$ex(n,\mathcal{Q}_{2})$ is the maximum size of a diamond-free family of
subsets of $[n]$. Despite of efforts made by many researchers, even the
asymptotic value of $ex(n,\mathcal{Q}_{2})$ has yet to be determined. It is
conjectured that

$ex(n,\mathcal{Q}_{2})=(2+o(1)){{n}\choose{\lfloor\frac{n}{2}\rfloor}}.$

Note that the two middle levels of the Boolean lattice do not contain a
diamond, we have

$ex(n,\mathcal{Q}_{2})\geq(2-o(1)){{n}\choose{\lfloor\frac{n}{2}\rfloor}}.$

Czabarka et al. [5] gave infinitely many asymptotically tight constructions by
using random set families defined from posets based on Abelian groups. Using
an elegant argument, Griggs et al. [20] showed that
$ex(n,\mathcal{Q}_{2})<2.296{{n}\choose{\lfloor\frac{n}{2}\rfloor}}$. This
bound was further improved to
$(2.25+o(1)){{n}\choose{\lfloor\frac{n}{2}\rfloor}}$ by Kramer et al. [27].
The best known upper bound on $ex(n,\mathcal{Q}_{2})$ is
$(2.20711+o(1)){{n}\choose{\lfloor\frac{n}{2}\rfloor}}$ provided by Grósz et
al. [21] as follows.

###### Theorem 3.3

(Grósz, Methuku, and Tompkins, [21]).

$ex(n,\mathcal{Q}_{2})\leq\bigg{(}\frac{\sqrt{2}+3}{2}+o(1)\bigg{)}{{n}\choose{\lfloor\frac{n}{2}\rfloor}}<(2.20711+o(1)){{n}\choose{\lfloor\frac{n}{2}\rfloor}}.$

In the proof of Theorem 3.3 which is Theorem 1.15 in [21], Grósz et al.
derived the next upper bound for the Lubell function $l(\mathcal{F})$.

###### Proposition 3.4

(Grósz, Methuku, and Tompkins, [21]). Suppose that $\mathcal{F}\subseteq
2^{[n]}$ is a family which contains no $\mathcal{Q}_{2}$. Then

$l(\mathcal{F})\leq\frac{\sqrt{2}+3}{2}+o(1)<2.20711+o(1).$

Clearly, Theorem 1.9 and Proposition 3.4 imply immediately the following
generalization of Proposition 3.4.

###### Proposition 3.5

Suppose that $\mathcal{V}$ is a family of subspaces of $\mathbb{F}_{q}^{n}$
which contains no $\mathcal{Q}_{2}$. Then

$l_{q}(\mathcal{F})\leq\frac{\sqrt{2}+3}{2}+o(1)<2.20711+o(1).$

From Proposition 3.5, the next generalization of Theorem 3.3 follows directly,
where $ex_{q}(n,\mathcal{Q}_{2})$ denotes the maximum size of a family of
subspaces of $\mathbb{F}_{q}^{n}$ which contains no diamond $\mathcal{Q}_{2}$.

###### Theorem 3.6

$ex_{q}(n,\mathcal{Q}_{2})\leq\bigg{(}\frac{\sqrt{2}+3}{2}+o(1)\bigg{)}\genfrac{[}{]}{0.0pt}{}{n}{\lfloor\frac{n}{2}\rfloor}<(2.20711+o(1))\genfrac{[}{]}{0.0pt}{}{n}{\lfloor\frac{n}{2}\rfloor}.$

Note that the two middle levels of the lattice $\mathcal{L}_{n}(q)$ of
subspaces of $\mathbb{F}_{q}^{n}$ do not contain a diamond $\mathcal{Q}_{2}$.
The maximum size $ex_{q}(n,\mathcal{Q}_{2})$ of a family of subspaces of
$\mathbb{F}_{q}^{n}$ containing no $\mathcal{Q}_{2}$ satisfies

$ex_{q}(n,\mathcal{Q}_{2})\geq(2-o(1))\genfrac{[}{]}{0.0pt}{}{n}{\lfloor\frac{n}{2}\rfloor}.$

We propose the following conjecture analogous to the corresponding one for
Boolean lattices.

###### Conjecture 3.7

The maximum size $ex_{q}(n,\mathcal{Q}_{2})$ of a family of subspaces of
$\mathbb{F}_{q}^{n}$ containing no diamond $\mathcal{Q}_{2}$ satisfies

$ex_{q}(n,\mathcal{Q}_{2})=2\genfrac{[}{]}{0.0pt}{}{n}{\lfloor\frac{n}{2}\rfloor}.$

###  3.3 Sharpening the LYM inequality

Note that

$l(\mathcal{F})=\sum_{F\in\mathcal{F}}\frac{1}{{{n}\choose{|F|}}}=\sum_{i=0}^{n}\frac{|\mathcal{F}_{i}|}{{{n}\choose{i}}}$

with each $\mathcal{F}_{i}=\mathcal{F}\cap{{[n]}\choose{i}}$. The next result
in [11] sharpens the LYM inequality by increasing the coefficients
$\frac{1}{{{n}\choose{i}}}$.

###### Theorem 3.8

(Erdős, Frankl, Kleitman, Saks, and Sz$\acute{e}$kely, [11]). Let
$\mathcal{F}\subseteq 2^{[n]}$ is a Sperner family (or antichain) and let $k$
be the smallest integer $t$ for which $\sum_{i\leq
t}\frac{|\mathcal{F}_{i}|}{{{n-1}\choose{i-1}}}>1$. Then

$\sum_{i<k}\frac{k}{i}\frac{|\mathcal{F}_{i}|}{{{n}\choose{i}}}+\sum_{i\geq
k}\frac{n-k}{n-i}\frac{|\mathcal{F}_{i}|}{{{n}\choose{i}}}\leq 1.$

By Theorems 1.11 and 3.8, we have the next generalization of Theorem 3.8
directly.

###### Theorem 3.9

Assume that $\mathcal{V}$ is a Sperner family of subspaces of
$\mathbb{F}_{q}^{n}$ and let $k$ be the smallest integer $t$ for which
$\sum_{i\leq t}\frac{|\mathcal{V}_{i}|}{\genfrac{[}{]}{0.0pt}{}{n-1}{i-1}}>1$,
where $\mathcal{V}_{i}=\mathcal{V}\cap\genfrac{[}{]}{0.0pt}{}{[n]}{i}$. Then

$\sum_{i<k}\frac{k}{i}\frac{|\mathcal{V}_{i}|}{\genfrac{[}{]}{0.0pt}{}{n}{i}}+\sum_{i\geq
k}\frac{n-k}{n-i}\frac{|\mathcal{V}_{i}|}{\genfrac{[}{]}{0.0pt}{}{n}{i}}\leq
1.$

Recently, Malec and Tompkins [30] proved the following strengthening of
Theorem 1.6. For a family $\mathcal{F}\subseteq 2^{[n]}$, let

$c(F)=\max\\{k\mid F\mbox{ participates in a }k\mbox{-chain consisting of sets
from }\mathcal{F}\\}.$

###### Theorem 3.10

(Malec and Tompkins, [30]) Let $\mathcal{F}\subseteq 2^{[n]}$ be an arbitrary
family of sets, then

$\sum_{F\in\mathcal{F}}\frac{1}{c(F){{n}\choose{|F|}}}\leq 1.$

Obviously, if a family $\mathcal{F}\subseteq 2^{[n]}$ is $k$-Sperner, then
$c(F)\leq k$ for every $F\in\mathcal{F}$, and so Theorem 3.10 implies Theorem
1.6. By applying Theorems 1.11 and 3.10, we obtain the following
generalization of Theorem 3.10, where

$c_{q}(V)=\max\\{k\mid V\mbox{ participates in a }k\mbox{-chain consisting of
subspaces from }\mathcal{V}\\}.$

###### Theorem 3.11

Let $\mathcal{V}$ be a family of subspaces of $\mathbb{F}_{q}^{n}$, then

$\sum_{V\in\mathcal{V}}\frac{1}{c_{q}(V)\genfrac{[}{]}{0.0pt}{}{n}{dim(V)}}\leq
1.$

##  4 Families with no s pairwise disjoint members

In this section, we will apply Theorem 1.9 to give proofs for Theorems 1.13
and 1.14. For non-triviality, we assume that $q\geq 2$.

###  4.1 Preliminaries

Given a family $\mathcal{F}\subseteq 2^{[n]}$, Frankl [12] calls the Lubell
function $l(\mathcal{F})$ the binomial norm, and denoted by

$||\mathcal{F}||=\sum_{F\in\mathcal{F}}\frac{1}{{{n}\choose{|F|}}}=l(\mathcal{F}).$

For $0\leq i\leq n$, set

$\mathcal{F}_{i}=\\{F\in\mathcal{F}\mid|F|=i\\}$

and define

$\varphi(i)=\frac{|\mathcal{F}_{i}|}{{{n}\choose{i}}}.$

Then

$||\mathcal{F}||=\sum_{i=0}^{n}\frac{|\mathcal{F}_{i}|}{{{n}\choose{i}}}=\sum_{i=0}^{n}\varphi(i).$

Define

$\varrho_{n}(\mathcal{F})=\frac{||\mathcal{F}||}{n+1}.$

Clearly, the binomial norm satisfies

$0\leq||\mathcal{F}||\leq n+1$

which implies that

$0\leq\varrho_{n}(\mathcal{F})\leq 1.$

###### Definition 4.1

The families $\mathcal{F}_{1},\mathcal{F}_{2},\dots,\mathcal{F}_{s}$ in
$2^{[n]}$ are called cross-dependent if there is no choice of
$F_{1}\in\mathcal{F}_{1},F_{2}\in\mathcal{F}_{2},\dots,F_{s}\in\mathcal{F}_{s}$
such that $F_{1},F_{2},\dots,F_{s}$ are pairwise disjoint.

The following theorem is Theorem 1.6 in [12].

###### Theorem 4.2

(Frankl, [12]). Suppose that
$\mathcal{F}_{1},\mathcal{F}_{2},\dots,\mathcal{F}_{s}\subseteq 2^{[n]}$ are
cross-dependent. Then

$\varrho_{n}(\mathcal{F}_{1})+\varrho_{n}(\mathcal{F}_{2})+\cdots+\varrho_{n}(\mathcal{F}_{s})\leq
s-1.$ (11)

Obviously, by the definition, (11) is equivalent to

$||\mathcal{F}_{1}||+||\mathcal{F}_{2}||+\cdots+||\mathcal{F}_{s}||\leq(s-1)(n+1).$
(12)

By taking $\mathcal{F}_{1}=\mathcal{F}_{2}=\dots=\mathcal{F}_{s}=\mathcal{F}$,
Theorem 4.2 has the next useful corollary.

###### Corollary 4.3

(Frankl, [12]). Suppose that $\mathcal{F}\subseteq 2^{[n]}$ contains no $s$
pairwise disjoint members. Then

$||\mathcal{F}||\leq\frac{(s-1)(n+1)}{s}.$

Recall that two subspaces $V_{1},V_{2}$ in the vector space
$\mathbb{F}_{q}^{n}$ are disjoint if $dim(V_{1}\cap V_{2})=0$.

###### Definition 4.4

The families $\mathcal{V}_{1},\mathcal{V}_{2},\dots,\mathcal{V}_{s}$ of
subspaces of $\mathbb{F}_{q}^{n}$ are called cross-dependent if there is no
choice of
$V_{1}\in\mathcal{V}_{1},V_{2}\in\mathcal{V}_{2},\dots,V_{s}\in\mathcal{V}_{s}$
such that $V_{1},V_{2},\dots,V_{s}$ are pairwise disjoint.

Let $\mathcal{V}$ be a family of subspaces of the vector space
$\mathbb{F}_{q}^{n}$. We define the $q$-binomial norm of $\mathcal{V}$ to be

$||\mathcal{V}||_{q}=l_{q}(\mathcal{V})=\sum_{V\in\mathcal{V}}\frac{1}{\genfrac{[}{]}{0.0pt}{}{n}{|V|}}.$

For $0\leq i\leq n$, set

$\mathcal{V}_{i}=\\{V\in\mathcal{V}\mid dim(V)=i\\}$

and define

$\varphi_{q}(i)=\frac{|\mathcal{V}_{i}|}{\genfrac{[}{]}{0.0pt}{}{n}{i}}.$

Then

$||\mathcal{V}||_{q}=\sum_{i=0}^{n}\frac{|\mathcal{V}_{i}|}{\genfrac{[}{]}{0.0pt}{}{n}{i}}=\sum_{i=0}^{n}\varphi_{q}(i).$
(13)

By Theorems 1.9 and 4.2, Corollary 4.3, and (12), we obtain the following
generalizations of Theorem 4.2 and Corollary 4.3 immediately.

###### Theorem 4.5

Suppose that families $\mathcal{V}_{1},\mathcal{V}_{2},\dots,\mathcal{V}_{s}$
of subspaces of $\mathbb{F}_{q}^{n}$ are cross-dependent. Then

$||\mathcal{V}_{1}||_{q}+||\mathcal{V}_{2}||_{q}+\cdots+||\mathcal{V}_{s}||_{q}\leq(s-1)(n+1).$

###### Theorem 4.6

Suppose that $\mathcal{V}$ is a family of subspaces of $\mathbb{F}_{q}^{n}$
which contains no $s$ pairwise disjoint members. Then

$||\mathcal{V}||_{q}\leq\frac{(s-1)(n+1)}{s}.$ (14)

###  4.2 Proofs of Theorems 1.13 and 1.14

Next, we derive Theorems 1.13 and 1.14 by applying Theorems 4.5 and 4.6,
following an approach similar to that of Frankl [12], but the analysis is more
complicated. Theorem 1.9 plays a crucial role in deriving Theorems 4.5 and
4.6.

The shadow of a family of subsets is extended naturally to vector spaces as
follows: For a family $\mathcal{V}\subseteq\genfrac{[}{]}{0.0pt}{}{[n]}{k}$,
the shadow of $\mathcal{V}$ is defined to be

$\partial\mathcal{V}=\bigg{\\{}W\in\genfrac{[}{]}{0.0pt}{}{[n]}{k-1}\bigg{|}W\subseteq
V\in\mathcal{V}\bigg{\\}}.$

The next $q$-analogue of Lov$\acute{a}$sz’s theorem for the Boolean lattices
is Theorem 1.4 in [4].

###### Theorem 4.7

(Chowdhury and Patkós, [4]) Let
$\mathcal{V}\subseteq\genfrac{[}{]}{0.0pt}{}{[n]}{k}$ and let $y$ be the real
number satisfying $|\mathcal{V}|=\genfrac{[}{]}{0.0pt}{}{y}{k}$. Then

$|\partial\mathcal{V}|\geq\genfrac{[}{]}{0.0pt}{}{y}{k-1}.$

A family $\mathcal{V}$ of subspaces of $\mathbb{F}_{q}^{n}$ is said to be a
complex if $W\subseteq V\in\mathcal{V}$ implies $W\in\mathcal{V}$. We have the
next lemma.

###### Lemma 4.8

Let a family $\mathcal{V}$ of subspaces of $\mathbb{F}_{q}^{n}$ be a complex
and let $\mathcal{V}_{i}=\\{V\in\mathcal{V}\mid dim(V)=i\\}$ for $0\leq i\leq
n$. Then the function
$\varphi_{q}(i)=\frac{|\mathcal{V}_{i}|}{\genfrac{[}{]}{0.0pt}{}{n}{i}}$ is
monotonic decreasing, i.e.,

$\varphi_{q}(0)\geq\varphi_{q}(1)\geq\cdots\geq\varphi_{q}(n).$

Proof. Note that

$\genfrac{[}{]}{0.0pt}{}{y}{k-1}=\frac{q^{k}-1}{q^{y-k+1}-1}\genfrac{[}{]}{0.0pt}{}{y}{k},$
$\genfrac{[}{]}{0.0pt}{}{n}{k-1}=\frac{q^{k}-1}{q^{n-k+1}-1}\genfrac{[}{]}{0.0pt}{}{n}{k}.$

It follows that for any $y\leq n$,

$\frac{\genfrac{[}{]}{0.0pt}{}{y}{k}}{\genfrac{[}{]}{0.0pt}{}{n}{k}}\leq\frac{\genfrac{[}{]}{0.0pt}{}{y}{k-1}}{\genfrac{[}{]}{0.0pt}{}{n}{k-1}}.$

Let $1\leq k\leq n$ be fixed and let $y$ be the real number satisfying
$|\mathcal{V}_{k}|=\genfrac{[}{]}{0.0pt}{}{y}{k}$. Then $y\leq n$. By Theorem
4.7,

$|\partial\mathcal{V}_{k}|\geq\genfrac{[}{]}{0.0pt}{}{y}{k-1}.$

Since $\mathcal{V}$ is a complex, we have
$\partial\mathcal{V}_{k}\subseteq\mathcal{V}_{k-1}$. It follows that

$\varphi_{q}(k-1)=\frac{|\mathcal{V}_{k-1}|}{\genfrac{[}{]}{0.0pt}{}{n}{k-1}}\geq\frac{|\partial\mathcal{V}_{k}|}{\genfrac{[}{]}{0.0pt}{}{n}{k-1}}\geq\frac{\genfrac{[}{]}{0.0pt}{}{y}{k-1}}{\genfrac{[}{]}{0.0pt}{}{n}{k-1}}\geq\frac{\genfrac{[}{]}{0.0pt}{}{y}{k}}{\genfrac{[}{]}{0.0pt}{}{n}{k}}=\frac{|\mathcal{V}_{k}|}{\genfrac{[}{]}{0.0pt}{}{n}{k}}=\varphi_{q}(k).$

$\hfill\Box$

Note that $\genfrac{[}{]}{0.0pt}{}{n}{k}$ defined by (1) satisfies the
symmetric and unimodal properties similar to those of the binomial
coefficients ${{n}\choose{k}}$. We have the following fact.

###### Lemma 4.9

Let $n,k,l,q$ be positive integers such that $l<k\leq n-1$, $q\geq 2$, and
$n\geq 3l$. Then

$\sum_{l\leq j\leq
k}\genfrac{[}{]}{0.0pt}{}{n}{j}\geq(k-l+1)\genfrac{[}{]}{0.0pt}{}{n}{l}.$ (15)

Proof. By the symmetric and unimodal properties of
$\genfrac{[}{]}{0.0pt}{}{n}{i}$, we have
$\genfrac{[}{]}{0.0pt}{}{n}{l}<\genfrac{[}{]}{0.0pt}{}{n}{j}$ for $l<j<n-l$.
Thus, (15) is obviously true for $k\leq n-l$, and so the lemma holds for
$l=0,1$. Hence we assume $l\geq 2$ and $k>n-l$.

Observe that

$\genfrac{[}{]}{0.0pt}{}{n}{l+1}=\frac{q^{n-l}-1}{q^{l+1}-1}\genfrac{[}{]}{0.0pt}{}{n}{l},$
(16)
$\genfrac{[}{]}{0.0pt}{}{n}{l+2}=\frac{q^{n-l-1}-1}{q^{l+2}-1}\genfrac{[}{]}{0.0pt}{}{n}{l+1}.$
(17)

We claim that $\genfrac{[}{]}{0.0pt}{}{n}{l+1}\geq
2\genfrac{[}{]}{0.0pt}{}{n}{l}$ and $\genfrac{[}{]}{0.0pt}{}{n}{l+2}\geq
2\genfrac{[}{]}{0.0pt}{}{n}{l}$ when $q\geq 2$ and $n\geq 3l$ except for
$n=3l$ and $l=2$. To see the claim, by (16) and (17), it suffices to show that
(i) $q^{n-l}-1\geq 2(q^{l+1}-1)$ and (ii) $q^{n-l-1}-1\geq q^{l+2}-1$. For
(ii), it is equivalent to $n-l-1\geq l+2$, i.e., $n\geq 2l+3$ which is true if
$n\geq 3l$ and $l\geq 3$ or $n\geq 3l+1$ and $l=2$. For (i), since $q\geq 2$
and $l\geq 2$, $n\geq 3l$ implies $n\geq 2l+2$ and so $q^{n-l}\geq q^{l+2}$
which implies that $q^{n-l}-1\geq q^{l+2}-1\geq 2q^{l+1}-1>2(2^{l+1}-1)$.
Thus, the claim holds.

We consider the following two cases:

Case 1. $n\neq 6$ when $l=2$. In this case, by the claim, we have

$\genfrac{[}{]}{0.0pt}{}{n}{l+1}+\genfrac{[}{]}{0.0pt}{}{n}{l+2}\geq
4\genfrac{[}{]}{0.0pt}{}{n}{l}.$ (18)

Since the terms $\genfrac{[}{]}{0.0pt}{}{n}{j}$ that exceed
$\genfrac{[}{]}{0.0pt}{}{n}{l}$ are $\genfrac{[}{]}{0.0pt}{}{n}{j}$ for
$l+1\leq j\leq n-l-1$ (there are $n-2l-1$ such terms) and
$\genfrac{[}{]}{0.0pt}{}{n}{l+1}$ and $\genfrac{[}{]}{0.0pt}{}{n}{l+2}$ are
the smallest, it follows (18) that

$\sum_{l+1\leq j\leq n-l-1}\genfrac{[}{]}{0.0pt}{}{n}{j}\geq
2(n-2l-1)\genfrac{[}{]}{0.0pt}{}{n}{l}.$ (19)

Adding $\genfrac{[}{]}{0.0pt}{}{n}{l}$ and $\genfrac{[}{]}{0.0pt}{}{n}{n-l}$
to (19) gives

$\sum_{l\leq j\leq n-l}\genfrac{[}{]}{0.0pt}{}{n}{j}\geq
2(n-2l)\genfrac{[}{]}{0.0pt}{}{n}{l}.$ (20)

Since $n\geq 3l$ and $k\leq n-1$, $2(n-2l)\geq k-l+1$. It follows from (20)
that

$\sum_{l\leq j\leq
n-l}\genfrac{[}{]}{0.0pt}{}{n}{j}\geq(k-l+1)\genfrac{[}{]}{0.0pt}{}{n}{l}.$
(21)

Clearly, (21) implies (15) when $k>n-l$.

Case 2. $n=6$ and $l=2$. Since $n-l<k\leq n-1$, we have $k=n-1=5$. To prove
(15), we need to verify

$\sum_{2\leq j\leq 5}\genfrac{[}{]}{0.0pt}{}{6}{j}\geq
4\genfrac{[}{]}{0.0pt}{}{6}{2}.$ (22)

By symmetry, $\genfrac{[}{]}{0.0pt}{}{6}{2}=\genfrac{[}{]}{0.0pt}{}{6}{4}$.
Hence (22) becomes

$\genfrac{[}{]}{0.0pt}{}{6}{3}+\genfrac{[}{]}{0.0pt}{}{6}{5}\geq
2\genfrac{[}{]}{0.0pt}{}{6}{2}.$ (23)

Since $q\geq 2$, $q^{3}\geq 2q^{2}\geq q^{2}+2q>q^{2}+q+1$. It follows from
(16) that

$\genfrac{[}{]}{0.0pt}{}{6}{3}=\frac{q^{4}-1}{q^{3}-1}\genfrac{[}{]}{0.0pt}{}{6}{2}=\frac{q^{3}+q^{2}+q+1}{q^{2}+q+1}\geq
2\genfrac{[}{]}{0.0pt}{}{6}{2}$

which implies (23). Therefore, the lemma holds. $\hfill\Box$

Recall that $\genfrac{[}{]}{0.0pt}{}{n}{k}$ is equal to the number of all
$k$-dimensional subspaces of $\mathbb{F}_{q}^{n}$. We have

$|\mathcal{L}_{n}(q)|=\sum_{i=0}^{n}\genfrac{[}{]}{0.0pt}{}{n}{i}$ (24)

which implies $||\mathcal{V}||_{q}\leq n+1$ for any
$\mathcal{V}\subseteq\mathcal{L}_{n}(q)$ (see (13)).

###### Definition 4.10

Fix a pair $(n,l)$ with $0\leq l<n$. We say $(n,l)$ is $q$-perfect if the
following inequality holds for all families $\mathcal{V}$ of subspaces of
$\mathbb{F}_{q}^{n}$ which are complexes with $||\mathcal{V}||_{q}<n+1$:

$|\mathcal{V}|\geq\sum_{0\leq
i<l}\genfrac{[}{]}{0.0pt}{}{n}{i}+(||\mathcal{V}||_{q}-l)\genfrac{[}{]}{0.0pt}{}{n}{l}.$
(25)

The next proposition is a $q$-analogue of Proposition 4.2 in [12], with a
similar proof by applying Lemmas 4.8 and 4.9 and replacing ${{n}\choose{i}}$’s
by $\genfrac{[}{]}{0.0pt}{}{n}{i}$’s (see Appendix 1 for a detailed proof).

###### Proposition 4.11

If $n\geq 3l$, then $(n,l)$ is $q$-perfect.

We are now ready to provide the following proofs for Theorems 1.13 and 1.14 by
applying Theorems 4.5 and 4.6 and Proposition 4.11.

Proof of Theorem 1.13. Note that if $\mathcal{V}$ contains no s pairwise
disjoint members, then the up-set

$\mathcal{V}^{*}=\\{W\in\mathcal{L}_{n}(q)\mid\exists V\in\mathcal{V},\mbox{
}V\subseteq W\\}$

has the same property as well. Thus, we may assume that $\mathcal{V}$ itself
is an up-set. Define

$\mathcal{F}=\mathcal{L}_{n}(q)\setminus\mathcal{V}.$

Then $\mathcal{F}$ is a complex as $\mathcal{V}$ is an up-set. By Theorem 4.6,
we have

$||\mathcal{V}||_{q}\leq\frac{(s-1)(n+1)}{s}$

which implies that $||\mathcal{F}||_{q}\geq\frac{n+1}{s}$ as
$||\mathcal{L}_{n}(q)||_{q}=n+1$ by (13) and (24).

For (i), we have $n=sk-1$ and so $||\mathcal{F}||_{q}\geq k$. Apply
Proposition 4.11 with $l=k-1$, we obtain

$|\mathcal{F}|\geq\sum_{0\leq
i<l}\genfrac{[}{]}{0.0pt}{}{n}{i}+(||\mathcal{F}||_{q}-l)\genfrac{[}{]}{0.0pt}{}{n}{l}\geq\sum_{0\leq
i\leq k-1}\genfrac{[}{]}{0.0pt}{}{n}{i}$

which gives (5).

For (ii), we have $n=sk+r$ with $0\leq r\leq s-2$ and so
$||\mathcal{F}||_{q}\geq k+\frac{r+1}{s}$. Apply Proposition 4.11 with $l=k$,
we obtain

$|\mathcal{F}|\geq\sum_{0\leq
i<k}\genfrac{[}{]}{0.0pt}{}{n}{i}+(||\mathcal{F}||_{q}-k)\genfrac{[}{]}{0.0pt}{}{n}{k}\geq\sum_{0\leq
i\leq
k-1}\genfrac{[}{]}{0.0pt}{}{n}{i}+\frac{r+1}{s}\genfrac{[}{]}{0.0pt}{}{n}{k}$

which gives (6).

For the sharpness in the case $n=sk-1$, take the family $\mathcal{V}$ of all
subspaces of dimension greater than or equal to $k$. Then
$|\mathcal{V}|=\sum_{i\geq k}\genfrac{[}{]}{0.0pt}{}{n}{i}$ and $\mathcal{V}$
contains no s pairwise disjoint members.

$\hfill\Box$

Proof of Theorem 1.14. Set
$\mathcal{W}_{i}=\mathcal{L}_{n}(q)\setminus\mathcal{V}_{i}$ for $1\leq i\leq
s$. By (24) and $n=sl+r$, (7) is equivalent to

$\sum_{1\leq i\leq s}|\mathcal{W}_{i}|\geq s\sum_{0\leq
j<l}\genfrac{[}{]}{0.0pt}{}{n}{j}+(r+1)\genfrac{[}{]}{0.0pt}{}{n}{l}.$ (26)

By Theorem 4.5, we obtain

$\sum_{1\leq i\leq s}||\mathcal{V}_{i}||_{q}\leq(s-1)(n+1).$

Note that $||\mathcal{W}_{i}||_{q}=(n+1)-||\mathcal{V}_{i}||_{q}$ for each
$1\leq i\leq s$. It follows that

$\sum_{1\leq i\leq s}||\mathcal{W}_{i}||_{q}\geq n+1.$

Since $(n,l)$ is $q$-perfect by Proposition 4.11, (25) implies that

$\sum_{1\leq i\leq s}|\mathcal{W}_{i}|\geq s\sum_{0\leq
j<l}\genfrac{[}{]}{0.0pt}{}{n}{j}+(n+1-sl)\genfrac{[}{]}{0.0pt}{}{n}{l},$

which gives (26).

For the sharpness of the bound in (7), consider the families

$\mathcal{V}_{1}=\cdots=\mathcal{V}_{r+1}=\\{V\in\mathcal{L}_{n}(q)\mid
dim(V)>l\\},$
$\mathcal{V}_{r+2}=\cdots=\mathcal{V}_{s}=\\{V\in\mathcal{L}_{n}(q)\mid
dim(V)\geq l\\}.$

This shows that the bound in (7) is best possible. $\hfill\Box$

##  5 Families without $d$-dimensional $q$-algebras in linear lattices

In this section, we apply Theorem 1.9 to derive generalizations of interesting
results about the maximum size of a family $\mathcal{F}\subseteq 2^{[n]}$
which does not contain a $d$-dimensional Boolean algebra. Recall that
$2^{[n]}$ is the power set of $[n]=\\{1,2,\dots,n\\}$. The following concept
of $d$-dimensional Boolean algebra is given in [22] and [24] (also called
$d$-dimensional combinatorial subspace in [33]).

###### Definition 5.1

A collection $\mathcal{B}\subseteq 2^{[n]}$ forms a $d$-dimensional Boolean
algebra if there exist pairwise disjoint sets
$X_{0},X_{1},\dots,X_{d}\subseteq[n]$, all non-empty with perhaps the
exception of $X_{0}$, so that

$\mathcal{B}=\bigg{\\{}X_{0}\cup(\cup_{i\in I}X_{i})\mid
I\subseteq[d]\bigg{\\}}.$

###### Definition 5.2

Given a positive integer d, define $b(n,d)$ to be the maximum size of a family
$\mathcal{F}\subseteq 2^{[n]}$ which does not contain a $d$-dimensional
Boolean algebra ($\mathcal{B}_{d}$-free).

Note that a $1$-dimensional Boolean algebra is simply a pair of sets, one
contained in the other and so, by Sperner’s theorem (Theorem 1.1),

$b(n,1)={{n}\choose{\lfloor\frac{n}{2}\rfloor}}\sim\sqrt{\frac{2}{\pi}}n^{-\frac{1}{2}}\cdot
2^{n}=o(2^{n}).$

Erdős and Kleitman [10] found that there exist constants $c_{1}$ and $c_{2}$
so that for $n$ sufficiently large,

$c_{1}n^{-\frac{1}{4}}\cdot 2^{n}\leq b(n,2)\leq c_{2}n^{-\frac{1}{4}}\cdot
2^{n}.$

In [22], the following bounds on $b(n,d)$ are proved by Gunderson et al.:

$n^{-\frac{(1+o(1))d}{2^{d+1}-2}}\cdot 2^{n}\leq b(n,d)\leq
10^{d}2^{-2^{1-d}}d^{d-2^{-d}}n^{-\frac{1}{2^{d}}}\cdot 2^{n}=o(2^{n}).$ (27)

The upper bound in (27) is improved by Johnston et al. [24] and Polymath [33]
as follows.

###### Theorem 5.3

(Polymath, [33]). The maximum size $b(n,d)$ of a family $\mathcal{F}\subseteq
2^{[n]}$ which does not contain a $d$-dimensional Boolean algebra satisfies

$b(n,d)\leq\bigg{(}\frac{25}{n}\bigg{)}^{\frac{1}{2^{d}}}\cdot 2^{n}.$

We remark that Sperner’s theorem and Theorem 5.3 were applied in a new proof
of Furstenberg and Katznelson’s density Hales-Jewett theorem (which implies
the well-known Szemerédi theorem, see Theorem 6.1 below) by Polymath [33].

###### Theorem 5.4

(Johnston, Lu, and Milans, [24]). There is a positive constant $C$,
independent of $d$, such that for every $d$ and all sufficiently large $n$,
the following is true.

$b(n,d)\leq Cn^{-\frac{1}{2^{d}}}\cdot 2^{n}.$

In order to prove Theorem 5.4, Johnston et al. [24] obtained the following
useful result as Corollary 1, where $l(\mathcal{F})$ is the Lubell function
defined in (2).

###### Theorem 5.5

(Johnston, Lu, and Milans, [24]). For $d\geq 3$ and
$n\geq(2^{d}-\frac{2}{ln2})^{2}$, every family $\mathcal{F}\subseteq 2^{[n]}$
containing no $d$-dimensional Boolean algebra satisfies

$l(\mathcal{F})\leq 2(n+1)^{1-2^{1-d}}.$

Similar to Definitions 5.1 and 5.2, one can define the corresponding concept
and value for vector spaces as follows. Recall that $\mathcal{L}_{n}(q)$
denotes the linear lattice of the set of all subspaces of
$\mathbb{F}_{q}^{n}$.

###### Definition 5.6

A collection $\mathcal{B}\subseteq\mathcal{L}_{n}(q)$ forms a $d$-dimensional
$q$-algebra if there exist pairwise disjoint sets
$X_{0},X_{1},\dots,X_{d}\subseteq\mathbb{F}_{q}^{n}$, all non-empty with
perhaps the exception of $X_{0}$, so that

$\mathcal{B}=\bigg{\\{}span\\{X_{0}\cup(\cup_{i\in I}X_{i})\\}\mid
I\subseteq[d]\bigg{\\}}.$

###### Definition 5.7

Given a positive integer d, define $b_{q}(n,d)$ to be the maximum size of a
family $\mathcal{V}\subseteq\mathcal{L}_{n}(q)$ which does not contain a
$d$-dimensional $q$-algebra.

Note that forbidden $d$-dimensional $q$-algebra is a hereditary property.
Theorems 1.9 and 5.5 together imply immediately the next $q$-analogue of
Theorem 5.5, where $l_{q}(\mathcal{V})$ is given in (4).

###### Theorem 5.8

For $d\geq 3$ and $n\geq(2^{d}-\frac{2}{ln2})^{2}$, every family
$\mathcal{V}\subseteq\mathcal{L}_{n}(q)$ containing no $d$-dimensional
$q$-algebra satisfies

$l_{q}(\mathcal{V})\leq 2(n+1)^{1-2^{1-d}}.$

Recall that $\genfrac{[}{]}{0.0pt}{}{n}{k}$ is symmetric and has the unimodal
property, we have
$\genfrac{[}{]}{0.0pt}{}{n}{k}\leq\genfrac{[}{]}{0.0pt}{}{n}{\lceil\frac{n}{2}\rceil}$
for all $k\leq n$. Theorem 5.8 gives the following immediate consequence which
is a generalization of Theorems 5.3 and 5.4.

###### Theorem 5.9

For $d\geq 3$ and $n\geq(2^{d}-\frac{2}{ln2})^{2}$, every family
$\mathcal{V}\subseteq\mathcal{L}_{n}(q)$ containing no $d$-dimensional
$q$-algebra satisfies

$|\mathcal{V}|\leq
2(n+1)^{1-2^{1-d}}\genfrac{[}{]}{0.0pt}{}{n}{\lceil\frac{n}{2}\rceil}.$

Gunderson et al. [22] and Johnston et al. [24] introduced the next Ramsey-type
concept: Given positive integers $n$ and $d$, define $r(d,n)$ to be the
largest integer $r$ so that every $r$-coloring of $2^{[n]}$ contains a
monochromatic copy of a $d$-dimensional Boolean algebra $\mathcal{B}_{d}$.
Similarly, one defines $r_{q}(d,n)$ to be the largest integer $r$ so that
every $r$-coloring of $\mathcal{L}_{n}(q)$ contains a monochromatic copy of a
$d$-dimensional $q$-algebra. The following bounds for $r(d,n)$ are given in
[22] and [24]:

$\bigg{\lfloor}\frac{1}{2}n^{\frac{2}{2^{d}}}\bigg{\rfloor}\leq r(d,n)\leq
n^{\frac{d}{2^{d}-1}(1+o(1))}.$

By applying Theorem 5.8, we can derive the next lower bound for $r_{q}(d,n)$.

###### Theorem 5.10

For $d\geq 3$ and $n\geq(2^{d}-\frac{2}{ln2})^{2}$, we have

$r_{q}(d,n)\geq\bigg{\lfloor}\frac{1}{2}n^{\frac{2}{2^{d}}}\bigg{\rfloor}.$

Proof. Let $r=\bigg{\lfloor}\frac{1}{2}n^{\frac{2}{2^{d}}}\bigg{\rfloor}$. For
every $r$-coloring of $\mathcal{L}_{n}(q)$ and $1\leq i\leq r$, let
$\mathcal{V}_{i}$ be the family of subspaces in color $i$. By the definition
of $l_{q}(\mathcal{V})$ (see (4)) and (24), we have

$\sum_{1\leq i\leq r}l_{q}(\mathcal{V}_{i})=l_{q}(\mathcal{L}_{n}(q))=n+1.$

It follows that there exists a color $1\leq i\leq r$ such that
$l_{q}(\mathcal{V}_{i})\geq\frac{n+1}{r}>2(n+1)^{1-2^{1-d}}$. By Theorem 5.8,
$\mathcal{V}_{i}$ contains a $d$-dimensional $q$-algebra. $\hfill\Box$

##  6 Concluding Remarks

We provided a general relationship for LYM inequalities between Boolean
lattice and linear lattice (Theorem 1.9) in section 2. As applications of
Theorem 1.9, we derived generalizations of some well-known theorems on LYM
inequalities and diamond-free families of subsets of $[n]$ in section 3; the
generalizations of the well-known Kleitman Theorem on matchings (Theorem 1.13)
and its generalization (Theorem 1.14) in section 4; and Theorems 5.8 and 5.9
on forbidden $d$-dimensional $q$-algebra in section 5.

In 1975, Szemerédi [37] proved the following famous theorem.

###### Theorem 6.1

(Szemerédi, [37]) For every positive integer $k$ and every $\delta>0$ there
exists $N$ such that every subset $A\subseteq[N]$ of size at least $\delta N$
contains an arithmetic progression of length $k$.

Szemerédi theorem has now three substantially different proofs and has several
generalitions, including the multidimensional Szemerédi theorem of Furstenberg
and Katznelson [16] (and a different proof was given by Gowers [18]). We
propose the following conjecture for a vector space variant of Szemerédi
theorem, where we say a family $\mathcal{V}$ of subspaces of the
$n$-dimensional vector space $\mathbb{F}_{q}^{n}$ contains an arithmetic
progression of length $k$ if $\mathcal{V}$ contains a subfamily
$\mathcal{V}^{\prime}$ such that the dimensions of the subspaces in
$\mathcal{V}^{\prime}$ form an arithmetic progression of length $k$.

###### Conjecture 6.2

For every positive integer $k$ and every $\delta>0$ there exists $n$ such that
every family $\mathcal{V}$ of subspaces of $\mathbb{F}_{q}^{n}$ with desity at
least $\delta$ contains an arithmetic progression of length $k$.

Recall that Sperner’s theorem and its multidimensional version (Theorem 5.3)
were applied in a new proof of Furstenberg and Katznelson’s density Hales-
Jewett theorem by Polymath [33]. We wonder if one could obtain a proof for
Conjecture 6.2 by following Polymath’s approach and applying Theorem 5.9 (a
generalization of Theorems 5.3).

Appendix 1

Proof of Proposition 4.11. Recall that
$\varphi_{q}(i)=\frac{|\mathcal{V}_{i}|}{\genfrac{[}{]}{0.0pt}{}{n}{i}}$ for
$0\leq i\leq n$. Note that since any family $\mathcal{V}$ of subspaces of
$\mathbb{F}_{q}^{n}$ is a complex, $\mathbb{F}_{q}^{n}\in\mathcal{V}$ implies
that $\mathcal{V}=\mathcal{L}_{n}(q)$ and so $||\mathcal{V}||_{q}=n+1$ by (13)
and (24). Hence we may assume
$\varphi_{q}(n)=\frac{|\mathcal{V}_{n}|}{\genfrac{[}{]}{0.0pt}{}{n}{n}}=0$ for
proving $q$-perfectness. Define

$\alpha(j)=\varphi_{q}(j)-\varphi_{q}(j+1)\mbox{ for }0\leq j\leq n-1.$

By Lemma 4.8, $\alpha(j)\geq 0$ for every $0\leq j\leq n-1$. Since
$\varphi_{q}(n)=0$, by the definition, we have

$\varphi_{q}(j)=\alpha(j)+\alpha(j+1)+\cdots+\alpha(n-1).$ (28)

Let $\mathcal{V}\subseteq\mathcal{L}_{n}(q)$ be a complex with
$||\mathcal{V}||_{q}<n+1$. Define

$\mathcal{V}^{+}=\\{V\in\mathcal{V}\mid dim(V)\geq l\\},$
$\mathcal{V}^{-}=\\{V\in\mathcal{V}\mid dim(V)<l\\}.$

Then $\mathcal{V}=\mathcal{V}^{+}\cup\mathcal{V}^{-}$. Multiplying both sides
of (15) by $\alpha(k)$ gives

$\sum_{l\leq j\leq
k}\alpha(k)\genfrac{[}{]}{0.0pt}{}{n}{j}\geq\alpha(k)(k-l+1)\genfrac{[}{]}{0.0pt}{}{n}{l}.$
(29)

By (28), we have

$||\mathcal{V}^{+}||_{q}=\sum_{l\leq j\leq n-1}\varphi_{q}(j)=\sum_{l\leq
j\leq n-1}\sum_{j\leq k\leq n-1}\alpha(k)=\sum_{l\leq k\leq
n-1}(k-l+1)\alpha(k).$ (30)

It follows from (29) and (30) that

$||\mathcal{V}^{+}||_{q}\genfrac{[}{]}{0.0pt}{}{n}{l}\leq\sum_{l\leq k\leq
n-1}\alpha(k)\sum_{l\leq j\leq k}\genfrac{[}{]}{0.0pt}{}{n}{j}\hskip
62.59605pt$ $=\sum_{l\leq j\leq n-1}\genfrac{[}{]}{0.0pt}{}{n}{j}\sum_{j\leq
k\leq n-1}\alpha(k)$ $=\sum_{l\leq j\leq
n-1}\genfrac{[}{]}{0.0pt}{}{n}{j}\varphi_{q}(j)\hskip 36.98857pt$
$=\sum_{l\leq j\leq n-1}|\mathcal{V}_{j}|=|\mathcal{V}^{+}|.\hskip 22.76219pt$
(31)

For $\mathcal{V}^{-}$, since $\varphi_{q}(j)\leq 1$ for each $0\leq j\leq n$,
$||\mathcal{V}^{-}||_{q}=\sum_{0\leq j<l}\varphi_{q}(j)$, and
$\genfrac{[}{]}{0.0pt}{}{n}{j}<\genfrac{[}{]}{0.0pt}{}{n}{l}$ for each $j<l$,
we have

$|\mathcal{V}^{-}|=\sum_{0\leq
j<l}\varphi_{q}(j)\genfrac{[}{]}{0.0pt}{}{n}{j}\hskip 108.12047pt$
$=\sum_{0\leq j<l}\genfrac{[}{]}{0.0pt}{}{n}{j}-\sum_{0\leq
j<l}(1-\varphi_{q}(j))\genfrac{[}{]}{0.0pt}{}{n}{j}$ $\geq\sum_{0\leq
j<l}\genfrac{[}{]}{0.0pt}{}{n}{j}-\sum_{0\leq
j<l}(1-\varphi_{q}(j))\genfrac{[}{]}{0.0pt}{}{n}{l}$ $=\sum_{0\leq
j<l}\genfrac{[}{]}{0.0pt}{}{n}{j}-(l-||\mathcal{V}^{-}||_{q})\genfrac{[}{]}{0.0pt}{}{n}{l}.\hskip
17.07164pt$ (32)

Observe that $|\mathcal{V}|=|\mathcal{V}^{+}|+|\mathcal{V}^{-}|$ and
$||\mathcal{V}||_{q}=||\mathcal{V}^{+}||_{q}+||\mathcal{V}^{-}||_{q}$. It
follows from (31) and (32) that

$|\mathcal{V}|\geq\sum_{0\leq
i<l}\genfrac{[}{]}{0.0pt}{}{n}{i}+(||\mathcal{V}||_{q}-l)\genfrac{[}{]}{0.0pt}{}{n}{l}$

which is (25). $\hfill\Box$

## Declaration of Competing Interest

The authors declare that they have no conflicts of interest to this work.

## Acknowledgement

The research is supported by the National Natural Science Foundation of China
(71973103, 11861019) and Guizhou Talent Development Project in Science and
Technology (KY[2018]046), Natural Science Foundation of Guizhou ([2019]1047,
[2020]1Z001, [2021]5609).

## References

  * [1] N. Alon, L. Babai, and H. Suzuki, Multilinear Polynomials and Frankl-Ray-Chaudhuri-Wilson Type Intersection Theorems, J. Combin. Theory, Ser. A, 58 (1991), 165-180.
  * [2] C. Bey, Polynomial LYM inequalities, Combinatorica, 25 (2005), 19-38.
  * [3] B. Bollobás, On generalized graphs, Acta Math. Acad. Sci. Hungar., 16 (1965), 447-452.
  * [4] A. Chowdhury and B. Patkós, Shadows and intersections in vector spaces, J. Combin. Theory, Ser. A, 117 (2010), 1095-1106.
  * [5] E. Czabarka, A. Dutle, T. Johnston, and L. A. Székely, Abelian groups yield many large families for the diamond problem, Eur. J. Math., 1 (2015), 320-328.
  * [6] K. Engel, Sperner Theory, Volume 65 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1997
  * [7] P. Erdős, On a lemma of Littlewood and Offord, Bulletin of the American Mathematical Society, 52 (1945), 898-902.
  * [8] P. Erdős, A problem on independent $r$-tuples, Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 8 (1965), 83-95.
  * [9] P. Erdős, C. Ko, and R. Rado, Intersection theorems for systems of finite sets, Q. J. Math. Oxford (2), 12 (1961), 313-320.
  * [10] P. Erdős and D. Kleitman, On collections of subsets containing no $4$-member Boolean algebra, Proc. Amer. Math. Soc., 28 (1971), 87-90.
  * [11] P. L. Erdős, P. Frankl, D. J. Kleitman, M. E. Saks, and L. A. Sz$\acute{e}$kely, Sharpening the LYM inequality, Combinatorica, 2 (1992), 287-293.
  * [12] P. Frankl, Extremal set theory for the binomial norm, J. Combin. Theory, Ser. A, 170 (2020), 105144.
  * [13] P. Frankl and R. Graham, Intersection theorems for vector spaces, Europ. J. Combin., 6 (1985), 183-187.
  * [14] P. Frankl and A. Kupavskii, Two problems on matchings in set families - in the footsteps of Erdős and Kleitman, J. Combin. Theory, Ser. B, 138 (2019), 286-313.
  * [15] P. Frankl and R. M. Wilson, Intersection theorems with geometric consequences, Combinatorica, 1 (1981), 357-368.
  * [16] H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math., 34 (1978), 275-291.
  * [17] D. Gerbner, The covering lemma and $q$-analogues of extremal set theory problems, Ars Mathematica Contemporanea, 2023, doi.org/10.26493/1855-3974.2677.b7f.
  * [18] W. T. Gowers, Hypergraph regularity and the multidimensional Szemerédi theorem, Annals of Mathematics, 166 (2007), 897-946.
  * [19] C. Greene and D. J. Kleitman, Proof techniques in the theory of finite sets, MAA Studies in Math., 17 (1978), 12-79.
  * [20] J. R. Griggs, W. Li, and L. Lu, Diamond-free families, J. Combin. Theory, Ser. A, 119 (2012), 310-322.
  * [21] D. Grósz, A. Methuku, and C. Tompkins, An upper bound on the size of diamond-free families of sets, J. Combin. Theory, Ser. A, 156 (2018), 164-194.
  * [22] D. S. Gunderson, V. Rödl, and A. Sidorenko, Extremal Problems for Sets Forming Boolean Algebras and Complete Partite Hypergraphs, J. Combin. Theory, Ser. A, 88 (1999), 342-367.
  * [23] W. N. Hsieh, Intersection theorems for systems of finite vector spaces, Discrete Math., 12 (1975), 1-16.
  * [24] T. Johnston, L. Lu and K. G. Milans, Boolean algebras and Lubell functions, J. Combin. Theory, Ser. A, 136 (2015), 174-183.
  * [25] D. A. Klain and G. C. Rota, A Continuous Analogue of Sperner’s Theorem, Communications on Pure and Applied Mathematics, L (1997), 205-223.
  * [26] D. J. Kleitman, Maximal number of subsets of a finite set no $k$ of which are pairwise disjoint, J. Combin. Theory, 5 (1968), 157-163.
  * [27] L. Kramer, R. R. Martin, and M. Young, On diamond-free subposets of the Boolean lattice, J. Combin. Theory, Ser. A, 120 (2013), 545-560.
  * [28] L. Lu and K. G. Milans, Set families with forbidden subposets, J. Combin. Theory, Ser. A, 136 (2015), 126-142.
  * [29] D. Lubell, A short proof of Sperner’s lemma, J. Combin. Theory, 1 (1966), 299.
  * [30] D. Malec and C. Tompkins, Localized versions of extremal problems, European Journal of Combinatorics, 112 (2023), 103715.
  * [31] A. M$\grave{e}$roueh, A LYM inequality for induced posets, J. Combin. Theory, Ser. A, 155 (2018), 398-417.
  * [32] L. D. Meshalkin, Generalization of Sperner’s theorem on the number of subsets of a finite set, Theory Probab. Appl., 8 (1963), 203-204.
  * [33] D. H. J. Polymath, A new proof of the density Hales-Jewett theorem, Annals of Mathematics, 175 (2012), 1283-1327.
  * [34] J. Qian and D. K. Ray-Chaudhuri, Extremal case of Frankl-Ray-Chaudhuri-Wilson inequality, J. Statist. Plann. Inference, 95 (2001), 293-306.
  * [35] D. K. Ray-Chaudhuri and R. M. Wilson, On $t$-designs, Osaca J. Math., 12 (1975), 737-744.
  * [36] W. Samotij, Subsets of Posets Minimising the Number of Chains, Transactions of the American Mathematical Society, 371 (2019), 7259-7274.
  * [37] E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica, 27 (1975), 199-245.
  * [38] K. Yamamoto, Logarithmic order of free distributive lattice, J. Math. Soc. Japan, 6 (1954), 343-353.