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##  Dissecting Self-Supervised Learning Methods for Surgical Computer Vision

###  ===== Supplementary Material =====

##  Appendix A Implementation details

###  i Training: Self-supervised pretraining

We use ResNet-50 [He et al., 2016] as the base encoder architecture and
3-layer multilayer perceptron (MLP) as the projection head for all the Self-
supervised methods. The projection head uses $2$ hidden layers ($2048$, $256$)
including ReLU activation and batch normalization with an input layer
dimension of size $2048$ and output layer dimension of size $4096$. Specific
implementation details for each method are as follows: MoCo v2 uses a queue of
size $65536$ with a decay parameter ($\lambda$) of $0.999$ and temperature
($\tau$) of $0.2$; SimCLR uses the temperature ($\tau$) of $0.1$; SwAV uses
$3$ Sinkhorm-Knopp iterations and regularization parameter of $0.05$; _DINO_
uses decay parameter ($\lambda$) of $0.996$, warm-up iterations $7500$, and
centering decay parameter $\lambda_{c}$ of $0.9$.

We conduct all the self-supervised training experiments on four V100 GPUs
using SGD optimizer with LARC [You et al., 2017] (“trust” coefficient $\eta$ =
$0.001$), a base learning rate of $0.1$, weight decay of $0.000001$, and
momentum of $0.9$. The batch size is set to $64$/GPU ($256$ total batch size)
except in the Batch size hyperparameter study where we use batches of size
$128$ ($32$/GPU), $256$ ($64$/GPU), $512$ ($128$/GPU) and $1024$ ($256$/GPU).
We use the VISSL framework121212https://github.com/facebookresearch/vissl to
run all the experiments using synchronized batch normalization [Peng et al.,
2018] and automatic mixed precision (AMP) [Micikevicius et al., 2017].

For the Initialization hyperparameter study, we initialize ResNet-50 weights
as follows: 1) fully-supervised Imagenet weights (“FS”) from
torchvision131313https://download.pytorch.org/models/resnet50-0676ba61.pth, 2)
self-supervised Imagenet weights (“SS”) for SimCLR from
VISSL141414https://dl.fbaipublicfiles.com/vissl/model_zoo/simclr_rn50_800ep_simclr_8node_resnet_16_07_20.7e8feed1/model_final_checkpoint_phase799.torch,
for MoCo v2 from their Github
repo151515https://dl.fbaipublicfiles.com/moco/moco_checkpoints/moco_v2_800ep/moco_v2_800ep_pretrain.pth.tar,
for SwAV from
VISSL161616https://dl.fbaipublicfiles.com/vissl/model_zoo/swav_in1k_rn50_800ep_swav_8node_resnet_27_07_20.a0a6b676/model_final_checkpoint_phase799.torch,
and for DINO from their Github
repo171717https://dl.fbaipublicfiles.com/dino/dino_resnet50_pretrain/dino_resnet50_pretrain.pth.

###  ii Training: Downstream tasks

Hyperparameter study: in all the experiments (Section. 3.3), after the self-
supervised training, the trunk of the encoder is frozen and the SSL head is
replaced by a linear head that maps the feature vector (2048) to the number of
outputs for the relevant task. The linear layer is trained on 1 GPU using an
SGD optimizer with a base learning rate of $3e-3$ and $1e-1$ for phase
recognition and tool presence detection, respectively. The layer is trained
for 30 epochs for phase with a step-wise learning rate decay of 0.3
(milestone: 15 epoch) and 50 epochs for tool with a step-wise learning rate
decay of 0.1 (milestones: 20, 30 and 40 epoch). Frames are sampled at 1 fps
for the linear evaluation described above.

Phase recognition is formulated as a multi-class classification problem where
a weighted cross-entropy loss is minimized. The class weights for phases is
computed using median frequency balancing [Eigen and Fergus, 2015] on the
training set.

Since tool presence detection is a multi-label classification problem, we
employ weighted binary cross-entropy loss:

$L=\sum^{C}_{c=1}\frac{-1}{N}(W_{c}y_{c}\log(\sigma(\hat{y}_{c}))+(1-y_{c})\log(\sigma(\hat{y}_{c}))),$
(5)

where $y_{c}$ is the ground truth tool presence label for class c,
$\hat{y}_{c}$ is the predicted probability for class c, and $W_{c}$ is the
class weight.

Data supply study: in the data supply study (Section 3.4), after the SSL
training, we follow a similar downstream training setup to the one mentioned
above, where we replace the SSL head with a linear layer. However, we train
the full model for downstream tasks without freezing the encoder’s trunk. For
the task of phase recognition, all the experiments are trained on 4 GPUs using
SGD optimizer with LARC [You et al., 2017] for 30 epochs. Further, we use
augmentations and train the model using a cross-entropy loss. When training
the temporal model, the TCN is trained on the features extracted after the
phase finetuning using an Adam optimizer [Kingma and Ba, 2014] for 100 epochs
with a base learning rate of $3e-3$ and a learning rate decay of $3e-1$
(milestone: 75 epoch). Frames are sampled at 1 fps for the linear finetuning
and TCN training described above.

Experiments involving tool presence detection, the models are trained on 4
GPUs using an Adam optimizer for 50 epochs with a base learning rate of $1e-5$
and learning rate decay of 0.1 (milestone: 25 epoch). As specified above, we
use weighted binary cross-entropy loss with the class weights computed using
inverse frequency balancing for different percentages and samples of the
training data used.

All the above training setting is employed when training models with 100% of
the labeled data. In lower labeled data settings, the number of epochs and
milestones are scaled inversely in proportion to the percentage of training
data. This provides the models with a similar number of updates irrespective
of the amount of labeled data available.

Generalization study: In this step of our study, presented in Section 3.5 and
4.4, we train all the models for the downstream tasks of phase recognition and
tool presence detection (HeiChole and CATARACTS) with the same setup described
previously in the ‘Data supply study’ paragraph. To comply with the evaluation
process of the HeiChole challenge, the TCN, with the fixed hyperparameters, is
trained on features extracted at ‘native’ fps of the challenge.

We follow the data pipeline introduced in Nwoye et al. [2022b] where we resize
the frame to $448\times 256$ resolution and apply horizontal flipping and
brightness/contrast shift as data augmentation strategies. Hyperparameters for
finetuning setup involving RDV head and linear head utilize SGD optimizer with
a batch size of $32$ and weight decay $1e-6$. The learning rate for the
backbone is kept at $1e-4$ whereas the RDV head and linear head use $1e-2$ as
the learning rate. For RDV head we utilize a mixture of linear and exponential
learning rate schedulers and train for $100$ epochs with early stopping,
whereas for linear head setup, we use a multi-step learning rate scheduler
with training epochs set to $40$. All the experiments are run using a single
NVIDIA A100 GPU.

On both the Endoscapes and CaDIS datasets, we train the downstream semantic
segmentation models with an additional DeepLabv3+ head on top of the encoder
(ResNet50). In all these experiments, the model is trained on images resized
to $480\times 270$ for 25 epochs with a batch size of 32 and a learning rate
of $3e-4$ on a single NVIDIA V100 GPU.

##  Appendix B Augmentation details

Table 10 shows the details of all the augmentations used for the self-
supervised pretraining and supervised finetuning experiments.

Table 10: Different image augmentations used in the self-supervised pretraining and supervised finetuning experiments. The Source “RA” uses the RandAugment [Cubuk et al., 2020b] implementations that randomly selects two augmentations from the list and applies the augmentation with a given probability ($prob$), magnitude ($M$), and standard deviation on the magnitude ($M_{\sigma}$). The source “torchvision” uses the torchvision implementation for Random-erasing, Horizontal-flip, and Multi-crop augmentations. | Name | Source | Parameters
---|---|---|---
Color | Sharpness | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
Brightness | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
Contrast | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
Color | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
Auto-contrast | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
Equalize | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
Random-erasing | torchvision | $prob=0.8,scale=[0.02,0.1]$
Geometric | Rotate | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
Translate-x | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
Translate-y | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
Shear-x | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
Shear-y | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
Horizontal-flip | torchvision | $prob=0.5$
Strong Color | Posterize | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
Solarize | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
invertion | RA | $prob=0.5,M=8.0,M_{\sigma}=0.5$
Multi Crop | MC2: multi-crop 2 | torchvision |  ${224x224=2}$, $scale=[0.5,1]$
MC2: multi-crop 4 | torchvision |  $(224x224=2),(96x96=2)$, $scale=[0.5,1]$
MC2: multi-crop 8 | torchvision |  $(224x224=2),(96x96=6)$, $scale=[0.5,1]$


# Some new Bollobás-type inequalities

Erfei Yue

Abstract

A family of disjoint pairs of finite sets $\mathcal{P}=\\{(A_{i},B_{i})\mid
i\in[m]\\}$ is called a Bollobás system if $A_{i}\cap B_{j}\neq\emptyset$ for
every $i\neq j$, and a skew Bollobás system if $A_{i}\cap B_{j}\neq\emptyset$
for every $i<j$. Bollobás proved that for a Bollobás system, the inequality

$\sum_{i=1}^{m}\binom{|A_{i}|+|B_{i}|}{|A_{i}|}^{-1}\leqslant 1$

holds. Hegedüs and Frankl generalized this theorem to skew Bollobás systems
with the inequality

$\sum_{i=1}^{m}\binom{|A_{i}|+|B_{i}|}{|A_{i}|}^{-1}\leqslant 1+n,$

provided $A_{i},B_{i}\subseteq[n]$. In this paper, we improve this inequality
to

$\sum_{i=1}^{m}\left((1+|A_{i}|+|B_{i}|)\binom{|A_{i}|+|B_{i}|}{|A_{i}|}\right)^{-1}\leqslant
1$

with probabilistic method. We also generalize this result to partitions of
sets on both symmetric and skew cases.

##  1 Introduction

Let $[n]=\\{1,\ldots,n\\}$ be an $n$-element set. A family $\mathcal{F}$
consisting of some subsets of $[n]$ is the main object of the research on
extremal set theory. The following concepts of Bollobás system and skew
Bollobás system were introduced by Bollobás [2] and Frankl [1, 3]
respectively.

######  Definition 1.1.

Suppose $\mathcal{P}=\\{(A_{i},B_{i})\mid i\in[m]\\}$ is a family of pairs of
sets, where $A_{i},B_{i}\subseteq[n]$. Then $\mathcal{P}$ is called a
_Bollobás system_ if

$A_{i}\cap B_{j}=\emptyset\Leftrightarrow i=j;$

and a _skew Bollobás system_ if

(1)

$A_{i}\cap B_{i}=\emptyset$,

(2)

$A_{i}\cap B_{j}\neq\emptyset$ when $i<j$.

To solve a problem on hypergraphs, Bollobás [2] proved the following theorem
in 1965. In 1975, Tarján [19] gave a more elegant proof for it independently.

######  Theorem 1.2 (Bollobás [2]).

Let $\mathcal{P}=\\{(A_{i},B_{i})\mid i\in[m]\\}$ be a Bollobás system, where
$A_{i},B_{i}\subseteq[n]$. Then

$\sum_{i=1}^{m}\frac{1}{\binom{|A_{i}|+|B_{i}|}{|A_{i}|}}\leqslant 1.$

The following uniform version is a corollary of Theorem 1.2 for $|A_{i}|=a$
and $|B_{i}|=b$, and it was proved by Jaeger and Payan [7] in 1971 and Katona
[9] in 1974 independently.

######  Corollary 1.3.

Let $\mathcal{P}=\\{(A_{i},B_{i})\mid i\in[m]\\}$ be a Bollobás system, where
$A_{i},B_{i}\subseteq[n]$, and $|A_{i}|=a,|B_{i}|=b$ for any $i$. Then
$m\leqslant\binom{a+b}{a}$.

Theorem 1.2 and 1.3 play a pivotal role in extremal set theory. They has
spurred significant research interest, leading to generalizations for vector
spaces by Lovász [11, 12, 13] in 1977 and affine spaces by Hegedüs [5] in
2015.

In 1982, using a creative polynomial method, Frankl [3] proved that the
inequality of Corollary 1.3 remains true if $\mathcal{P}$ is a skew Bollobás
system (instead of Bollobás system). However, the condition of Theorem 1.2
cannot be weakened in the same way. So it is natural to ask what can we say
for a (nonuniform) skew Bollobás system. In 2023, Hegedüs and Frankl [6]
answered the question with the following theorem.

######  Theorem 1.4 (Hegedüs and Frankl [6]).

Suppose $A_{i},B_{i}\subseteq[n],i\in[m]$, and
$\mathcal{P}=\\{(A_{i},B_{i})\mid i\in[m]\\}$ is a skew Bollobás system. Then

$\sum_{i=1}^{m}\frac{1}{\binom{|A_{i}|+|B_{i}|}{|A_{i}|}}\leqslant 1+n.$

Although the inequality in Theorem 1.4 is tight, we can still strengthen it in
the following way.

######  Theorem 1.5.

Suppose $A_{i},B_{i}\subseteq[n],i\in[m]$, and
$\mathcal{P}=\\{(A_{i},B_{i})\mid i\in[m]\\}$ is a skew Bollobás system. Then

$\sum_{i=1}^{m}\frac{1}{(1+|A_{i}|+|B_{i}|)\binom{|A_{i}|+|B_{i}|}{|A_{i}|}}\leqslant
1.$

In 1984, Furedi [4] investigated the threshold (or $t$-intersecting) case of
Bollobás-type theorem. Subsequent work by Zhu [23], Talbot [18], and Kang,
Kim, and Kim [8] further extended the results. Recently, more variations of
Bollobás-type theorem are studied, such as Király, Nagy, Pálvölgyi, and
Visontai [10], O’Neill and Verstraëte [16], Scott and Wilmer [17], and Yu,
Kong, Xi, Zhang, and Ge [21].

The uniform version of Bollobás Theorem (Theorem 1.3) was generalized to
$r$-partitions by Yue, Lv, and Wang [22]. More specifically, the following
theorem decides the maximum cardinality of a (skew) Bollobás system in which
every set has a fixed size in each part of the ground set.

######  Theorem 1.6 (Yue, Lv, and Wang [22]).

Suppose $X=[n]$ is the disjoint union of some sets $X_{1},\ldots,X_{r}$. For
some fixed $a_{k},b_{k}$, a skew Bollobás system
$\mathcal{P}=\\{(A_{i},B_{i})\mid i\in[m]\\}$ satisfy that

$A_{i},B_{i}\subseteq X,|A_{i}\cap X_{k}|=a_{k},|B_{i}\cap
X_{k}|=b_{k},\forall i\in[m],k\in[r].$

Then we have

$m\leqslant\prod_{k=1}^{r}\binom{a_{k}+b_{k}}{a_{k}}.$

In the following theorem, we generalize Theorem 1.6 to the nonuniform version.

######  Theorem 1.7.

Suppose $X=[n]$ is the disjoint union of some sets $X_{1},\ldots,X_{r}$, and
$|X_{k}|=n_{k}$. For some $A_{i},B_{i}\subseteq X$,
$\mathcal{P}=\\{(A_{i},B_{i})\mid i\in[m]\\}$ is a skew Bollobás system. Then

$\sum_{i=1}^{m}\left(\prod_{k=1}^{r}\binom{|A_{i}\cap X_{k}|+|B_{i}\cap
X_{k}|}{|A_{i}\cap
X_{k}|}\right)^{-1}\leqslant\prod_{k=1}^{r}(1+n_{k})\leqslant\left(1+\frac{n}{r}\right)^{r}.$
(1)

This theorem shows two things. Firstly, if we fixed a partition
$X=X_{1}\cup\cdots\cup X_{r}$, then the maximum value of the left hand side of
(1) is $\prod_{k=1}^{r}(1+n_{k})$. Secondly, if we only fixed the ground set
$X$, while the partition is variable, then the maximum value of it is
$\left(1+\frac{n}{r}\right)^{r}$, and it meets its maximum when each part of
the partition has the same size. The first inequality in (1) is tight for all
$n_{1},\ldots,n_{r}$, and the second inequality is also tight if $r|n$, and
$X_{k}$’s are divided equally. Note that the case $r=1$ is Theorem 1.4.

We can also consider the symmetric version of Theorem 1.7. Surprisingly, this
time the symmetric version is harder than the skew version. For the case
$r=2$, we can prove the following tight inequality for a Bollobás system.

######  Theorem 1.8.

Suppose $X=[n]$ is the disjoint union of some sets $X_{1}$ and $X_{2}$, and
for some $A_{i},B_{i}\subseteq X$, $\mathcal{P}=\\{(A_{i},B_{i})\mid
i\in[m]\\}$ is a Bollobás system. Then

$\sum_{i=1}^{m}\left(\binom{|A_{i}\cap X_{1}|+|B_{i}\cap X_{1}|}{|A_{i}\cap
X_{1}|}\binom{|A_{i}\cap X_{2}|+|B_{i}\cap X_{2}|}{|A_{i}\cap
X_{2}|}\right)^{-1}\leqslant 1+\left\lfloor\frac{n}{2}\right\rfloor.$

And for an arbitrary $r\geqslant 1$, we can prove the following theorem.

######  Theorem 1.9.

Suppose $X=[n]$ is the disjoint union of some sets $X_{1},\ldots,X_{r}$, and
for some $A_{i},B_{i}\subseteq X$, $\mathcal{P}=\\{(A_{i},B_{i})\mid
i\in[m]\\}$ is a Bollobás system. Then

$\sum_{i=1}^{m}\left(\prod_{k=1}^{r}\binom{|A_{i}\cap X_{k}|+|B_{i}\cap
X_{k}|}{|A_{i}\cap
X_{k}|}\right)^{-1}\leqslant\left(1+\frac{n}{r}\right)^{r-1}.$

The case $r=1$ is the original Bollobás Theorem (Theorem 1.2). And the case
$r=2$ leads to an inequality

$\sum_{i=1}^{m}\left(\binom{|A_{i}\cap X_{1}|+|B_{i}\cap X_{1}|}{|A_{i}\cap
X_{1}|}\binom{|A_{i}\cap X_{2}|+|B_{i}\cap X_{2}|}{|A_{i}\cap
X_{2}|}\right)^{-1}\leqslant 1+\frac{n}{2},$

which is slightly weaker than the one in Theorem 1.8, but still tight for even
$n$’s. For the case $r\geqslant 3$, unfortunately, the upper bound given in
Theorem 1.9 does not seem to be tight.

The paper is organized as follows. Section 2 focuses on proving Theorem 1.5
and Theorem 1.7. We also discuss the tightness of Theorem 1.7. Section 3
presents proofs for Theorem 1.8 and Theorem 1.9. Additionally, we leverage
these theorems to derive a generalized form of the LYM-inequality.

##  2 Proof of the skew version and its tightness

For a fixed (finite) set $X$, let $S(X)$ be the group consisting of all
permutations on $X$, and $S_{n}=S([n])$. For a permutation $\sigma\in S(X)$
and a subset $A\subseteq X$, denote $\sigma(A)$ the image of $A$. In other
words, $\sigma(A)=\\{\sigma(a)\mid a\in A\\}$. In this section, we will prove
Theorem 1.5 and Theorem 1.7 with a probabilistic method.

######  Theorem 1.5.

Suppose $A_{i},B_{i}\subseteq[n],i\in[m]$, and
$\mathcal{P}=\\{(A_{i},B_{i})\mid i\in[m]\\}$ is a skew Bollobás system. Then

$\sum_{i=1}^{m}\frac{1}{(1+|A_{i}|+|B_{i}|)\binom{|A_{i}|+|B_{i}|}{|A_{i}|}}\leqslant
1.$

###### Proof.

Pick a random permutation $\sigma\in S_{n+1}$ uniformly. Note that

$\mathcal{P}^{\prime}=\\{(\sigma(A_{i}),\sigma(B_{i}))\mid(A_{i},B_{i})\in\mathcal{P},i\in[m]\\}$

is still a skew Bollobás system. The order of elements in $[n+1]$ is
rearranged by $\sigma$. For any $i$, let $E_{i}$ be the event that all
elements of $A_{i}$ go before $B_{i}$, and they are separated by the extra
element $n+1$. In other words,

$E_{i}=\\{\sigma\in S_{n+1}\mid\sigma(a)<\sigma(n+1)<\sigma(b),\forall a\in
A_{i},b\in B_{i}\\}.$

Now we claim that $E_{i}$ and $E_{j}$ cannot happen at the same time if $i\neq
j$. Actually, suppose $\sigma\in E_{i}\cap E_{j}$. For any $a\in A_{i}$ and
$b\in B_{j}$, we have $\sigma(a)<\sigma(n+1)<\sigma(b)$. Hence
$\sigma(A_{i})\cap\sigma(B_{j})=\emptyset$. Similarly, we can prove that
$\sigma(A_{j})\cap\sigma(B_{i})=\emptyset$. That is a contradiction to the
fact that $\mathcal{P}^{\prime}$ is a skew Bollobás system. Then we have

$\mathbb{P}\left(\bigcup_{i=1}^{m}E_{i}\right)=\sum_{i=1}^{m}\mathbb{P}(E_{i}).$

| $\sigma(A_{i})$ |  |  |  |  | $\sigma(B_{i})$
---|---|---|---|---|---|---
|  |  |  | $\sigma(n+1)$ |  |  |  |
$\sigma(A_{j})$ |  |  |  |  | $\sigma(B_{j})$ |
A schematic diagram for the proof

To finish the proof, we need to calculate the probability of each $E_{i}$.
Note that we only need to consider the order of elements in $\sigma(A_{i}\cup
B_{i}\cup\\{n+1\\})$. Within the $1+|A_{i}|+|B_{i}|$ places, there is exactly
$1$ slot for $n+1$. And for the elements in $A_{i}\cup B_{i}$, the chance of
any $|A_{i}|$ elements that lead the series is equal. There are
$\binom{|A_{i}|+|B_{i}|}{|A_{i}|}$ different possibilities, and exactly $1$ of
them is what we need. Hence
$\mathbb{P}(E_{i})=\left((1+|A_{i}|+|B_{i}|)\binom{|A_{i}|+|B_{i}|}{|A_{i}|}\right)^{-1}$,
and so

$1\geqslant\mathbb{P}\left(\bigcup_{i=1}^{m}E_{i}\right)=\sum_{i=1}^{m}\frac{1}{(1+|A_{i}|+|B_{i}|)\binom{|A_{i}|+|B_{i}|}{|A_{i}|}}.$

∎

Using a similar idea, we can also prove Theorem 1.7, but the progress would be
more tricky. For a finite set $X=\\{x_{1},\ldots,x_{k}\\}$, where
$x_{1}<\cdots<x_{k}$, let $X^{+}=X\cup\\{x_{k}+\frac{1}{2}\\}$, and $S(X)$ be
the collection of all permutations on $X$.

######  Theorem 1.7.

Suppose $X=[n]$ is the disjoint union of some sets $X_{1},\ldots,X_{r}$, and
for some $A_{i},B_{i}\subseteq X$, $\mathcal{P}=\\{(A_{i},B_{i})\mid
i\in[m]\\}$ is a skew Bollobás system. Then

$\sum_{i=1}^{m}\left(\prod_{k=1}^{r}\binom{|A_{i}\cap X_{k}|+|B_{i}\cap
X_{k}|}{|A_{i}\cap
X_{k}|}\right)^{-1}\leqslant\prod_{k=1}^{r}(1+n_{k})\leqslant\left(1+\frac{n}{r}\right)^{r}.$

###### Proof.

Without loss of generality, suppose
$X_{1}=\\{1,\ldots,n_{1}\\},X_{2}=\\{n_{1}+1,\ldots,n_{1}+n_{2}\\},\ldots,X_{r}=\\{n_{1}+\cdots+n_{r-1}+1,\ldots,n_{1}+\cdots+n_{r}\\}$,
where $n_{1}+\cdots+n_{r}=n$. Let $X^{\prime}=X_{1}^{+}\cup\cdots\cup
X_{r}^{+}$, and

$\Omega=\\{\sigma\in S(X^{\prime})\mid\sigma(X_{k}^{+})=X_{k}^{+},\forall
k\in[r]\\}.$

Pick a random permutation $\sigma\in\Omega$ uniformly. Then

$\mathcal{P}^{\prime}=\\{(\sigma(A_{i}),\sigma(B_{i}))\mid(A_{i},B_{i})\in\mathcal{P}\\}$

is still a skew Bollobás system. For any $i\in[m]$ and $k\in[r]$, let

$E_{ik}=\\{\sigma\in\Omega\mid\sigma(a)<\sigma(n_{1}+\cdots+n_{k}+1/2)<\sigma(b),\forall
a\in A_{i}\cap X_{k},b\in B_{i}\cap X_{k}\\},$

and

$E_{i}=\bigcap_{k=1}^{r}E_{ik}.$

Note that for a fixed $i$, the events $E_{ik}$ are independent. Using this
fact and the similar argument adopted in the proof of Theorem 1.5, we have

$\begin{split}\mathbb{P}(E_{i})&=\prod_{k=1}^{r}\mathbb{P}(E_{ik})=\prod_{k=1}^{r}\left((1+|A_{i}\cap
X_{k}|+|B_{i}\cap X_{k}|)\binom{|A_{i}\cap X_{k}|+|B_{i}\cap
X_{k}|}{|A_{i}\cap X_{k}|}\right)^{-1}\\\
&\geqslant\prod_{k=1}^{r}\left((1+n_{k})\binom{|A_{i}\cap X_{k}|+|B_{i}\cap
X_{k}|}{|A_{i}\cap X_{k}|}\right)^{-1}.\end{split}$

Now we claim $E_{i}\cap E_{j}=\emptyset$ for distinct $i,j$. Suppose for the
contradiction that $\sigma\in E_{i}\cap E_{j}$. Then for any $k$, we have
$\sigma\in E_{ik}\cap E_{jk}$,so
$\sigma(a)<\sigma(n_{1}+\cdots+n_{k}+\frac{1}{2})<\sigma(b)$ for any $a\in
A_{i}\cap X_{k},b\in B_{j}\cap X_{k}$. Then $\sigma(A_{i}\cap X_{k})$ and
$\sigma(B_{j}\cap X_{k})$ are disjoint. Similarly, we can prove that
$\sigma(A_{j}\cap X_{k})$ and $\sigma(B_{i}\cap X_{k})$ are disjoint. Hence
$\sigma(A_{i})\cap\sigma(B_{j})=\sigma(A_{j})\cap\sigma(B_{i})=\emptyset$,
which contradicts the fact that $\mathcal{P}^{\prime}$ is a skew Bollobás
system. In this way, we have

$1\geqslant\mathbb{P}(\bigcup_{i=1}^{m}E_{i})=\sum_{i=1}^{m}\mathbb{P}(E_{i})\geqslant\sum_{i=1}^{m}\prod_{k=1}^{r}\left((1+n_{k})\binom{|A_{i}\cap
X_{k}|+|B_{i}\cap X_{k}|}{|A_{i}\cap X_{k}|}\right)^{-1},$

so

$\sum_{i=1}^{m}\left(\prod_{k=1}^{r}\binom{|A_{i}\cap X_{k}|+|B_{i}\cap
X_{k}|}{|A_{i}\cap
X_{k}|}\right)^{-1}\leqslant\prod_{k=1}^{r}(1+n_{k})\leqslant\left(\frac{1}{r}\sum_{k=1}^{r}(1+n_{k})\right)^{r}=\left(1+\frac{n}{r}\right)^{r}.$

∎

The next example shows the tightness of Theorem 1.7.

######  Example 1.

Suppose $X=[n]$ is the disjoint union of $X_{1},\ldots,X_{r}$, where
$|X_{k}|=n_{k}$. Let $A_{1},\ldots,A_{2^{n}}$ be all subsets of $X$, and
$|A_{1}|\geqslant\cdots\geqslant|A_{2^{n}}|$. Then
$\mathcal{P}=\\{(A_{i},[n]\setminus A_{i})\mid i\in[2^{n}]\\}$ is a skew
Bollobás system. For a fixed $i$, let $|A_{i}\cap X_{k}|=a_{k}$, and
$(a_{1},\ldots,a_{r})$ is called the type of $A_{i}$. Then the contribution of
$A_{i}$ to the sum is $\prod_{k=1}^{r}\binom{n_{k}}{a_{k}}^{-1}$. On the other
hand, there are altogether $\prod_{k=1}^{r}\binom{n_{k}}{a_{k}}$ many of
$A_{j}$’s that have the same type with $A_{i}$. So the total contribution of a
type of $A_{i}$’s to the sum is exactly $1$. Note that there are
$\prod_{k=1}^{r}(1+n_{k})$ many of different types, because
$a_{k}\in\\{0,1,\ldots,n_{k}\\}$, and each $a_{k}$’s are valued independently.
Hence

$\sum_{i=1}^{m}\left(\prod_{k=1}^{r}\binom{|A_{i}\cap X_{k}|+|B_{i}\cap
X_{k}|}{|A_{i}\cap X_{k}|}\right)^{-1}=\prod_{k=1}^{r}(1+n_{k}),$

and so the first inequality of Theorem 1.7 is tight for any $n$ and $r$. In
particular, if $r|n$, and $n_{1}=\cdots=n_{r}=\frac{n}{r}$, the second
inequality is also tight.

##  3 Proof of the symmetric version and an application

In this section, we will prove Theorem 1.8 and Theorem 1.9, and show an
application of them. Firstly, we begin this work with proving the following
lemma.

######  Lemma 3.1.

Suppose $X=[n]$ is the disjoint union of some sets $X_{1},\ldots,X_{r}$, and
$|X_{k}|=n_{k}$. For some $A_{i},B_{i}\subseteq X$,
$\mathcal{P}=\\{(A_{i},B_{i})\mid i\in[m]\\}$ is a Bollobás system. Then for
any $l\in[r]$, we have

$\sum_{i=1}^{m}\left(\prod_{k=1}^{r}\binom{|A_{i}\cap X_{k}|+|B_{i}\cap
X_{k}|}{|A_{i}\cap
X_{k}|}\right)^{-1}\leqslant\frac{\prod_{k=1}^{r}(1+n_{k})}{1+n_{l}}.$ (2)

###### Proof.

Without loss of generality, we may assume that
$X_{1}=\\{1,\ldots,n_{1}\\},X_{2}=\\{n_{1}+1,\ldots,n_{1}+n_{2}\\},\ldots,X_{r}=\\{n_{1}+\cdots+n_{r-1}+1,\ldots,n_{1}+\cdots+n_{r}\\}$.
For a fixed $l$, denote

$X_{k}^{\prime}=\begin{cases}X_{k}^{+},&\textrm{if~{}}k\neq l,\\\
X_{k},&\textrm{if~{}}k=l,\end{cases}$

and $X^{\prime}=\bigcup_{k=1}^{r}X_{k}^{\prime}$. Let

$\Omega=\\{\sigma\in
S(X^{\prime})\mid\sigma(X_{k}^{\prime})=X_{k}^{\prime},\forall k\in[r]\\}.$

Pick a random permutation $\sigma\in\Omega$ uniformly. Then

$\mathcal{P}^{\prime}=\\{(\sigma(A_{i}),\sigma(B_{i}))\mid(A_{i},B_{i})\in\mathcal{P}\\}$

is still a Bollobás system. For any $i\in[m]$ and $k\neq l$, let

$\displaystyle E_{il}$ $\displaystyle=$
$\displaystyle\\{\sigma\in\Omega\mid\sigma(a)<\sigma(b),\forall a\in A_{i}\cap
X_{l},b\in B_{i}\cap X_{l}\\},$ $\displaystyle E_{ik}$ $\displaystyle=$
$\displaystyle\\{\sigma\in\Omega\mid\sigma(a)<\sigma(n_{1}+\cdots+n_{k}+1/2)<\sigma(b),\forall
a\in A_{i}\cap X_{k},b\in B_{i}\cap X_{k}\\},$

and

$E_{i}=\bigcap_{k=1}^{r}E_{ik}.$

Note that for a fixed $i$, the events $E_{ik}$’s are independent, and we have

$\mathbb{P}(E_{il})=\binom{|A_{i}\cap X_{k}|+|B_{i}\cap X_{k}|}{|A_{i}\cap
X_{k}|}^{-1},$

and

$\begin{split}\mathbb{P}(E_{ik})&=\left((1+|A_{i}\cap X_{k}|+|B_{i}\cap
X_{k}|)\binom{|A_{i}\cap X_{k}|+|B_{i}\cap X_{k}|}{|A_{i}\cap
X_{k}|}\right)^{-1}\\\ &\geqslant\left((1+n_{k})\binom{|A_{i}\cap
X_{k}|+|B_{i}\cap X_{k}|}{|A_{i}\cap X_{k}|}\right)^{-1},\end{split}$

for any $k\neq l$, so

$\mathbb{P}(E_{i})\geqslant(n_{l}+1)\prod_{k=1}^{r}\left((n_{k}+1)\binom{|A_{i}\cap
X_{k}|+|B_{i}\cap X_{k}|}{|A_{i}\cap X_{k}|}\right)^{-1}.$

Now we claim that $E_{i}\cap E_{j}=\emptyset$ for distinct $i,j$. Suppose on
the contrary there is a $\sigma\in E_{i}\cap E_{j}$. Then like the discussion
in the proof of Theorem 1.7, we can prove that $\sigma(A_{i}\cap X_{k})$ and
$\sigma(B_{j}\cap X_{k})$ are disjoint for any $k\neq l$, as well as
$\sigma(A_{j}\cap X_{k})$ and $\sigma(B_{i}\cap X_{k})$. Suppose
$x\in\sigma(A_{i}\cap X_{l})\cap\sigma(B_{j}\cap X_{l})$. Then we have
$\sigma(a)<x<\sigma(b)$ for any $a\in A_{j}\cap X_{l}$ and $b\in B_{i}\cap
X_{l}$, so $\sigma(A_{j}\cap X_{l})$ and $\sigma(B_{i}\cap X_{l})$ are
disjoint, which contradicts the fact that $\mathcal{P}^{\prime}$ is a Bollobás
system.

| $\sigma(A_{i}\cap X_{l})$ |  | $\sigma(B_{i}\cap X_{l})$
---|---|---|---
|  |  |  | $x$ |  |  |  |
$\sigma(A_{j}\cap X_{l})$ |  | $\sigma(B_{j}\cap X_{l})$ |
A schematic diagram for the proof

Then we have

$1\geqslant\mathbb{P}(\bigcup_{i=1}^{m}E_{i})\geqslant\sum_{i=1}^{m}(1+n_{l})\prod_{k=1}^{r}\left((1+n_{k})\binom{|A_{i}\cap
X_{k}|+|B_{i}\cap X_{k}|}{|A_{i}\cap X_{k}|}\right)^{-1},$

and so

$\sum_{i=1}^{m}\prod_{k=1}^{r}\binom{|A_{i}\cap X_{k}|+|B_{i}\cap
X_{k}|}{|A_{i}\cap
X_{k}|}^{-1}\leqslant\frac{\prod_{k=1}^{r}(1+n_{k})}{1+n_{l}}.$

∎

For the case $r=2$, Lemma 3.1 implies

$\sum_{i=1}^{m}\ \left(\binom{|A_{i}\cap X_{1}|+|B_{i}\cap X_{1}|}{|A_{i}\cap
X_{1}|}\binom{|A_{i}\cap X_{2}|+|B_{i}\cap X_{2}|}{|A_{i}\cap
X_{2}|}\right)^{-1}\leqslant\min\\{1+n_{1},1+n_{2}\\}\leqslant
1+\left\lfloor\frac{n}{2}\right\rfloor,$

which is the result of Theorem 1.8. The following example proves the tightness
of it.

######  Example 2.

Suppose $k=\lfloor\frac{n}{2}\rfloor,m=\binom{n}{k}$, and
$X=[n],X_{1}=[k],X_{2}=X\setminus X_{1}$. Let
$\binom{X}{k}=\\{A_{1},\ldots,A_{m}\\}$, and $B_{i}=[n]\setminus A_{i}$. Then
$\mathcal{P}=\\{(A_{i},B_{i})\mid i\in[m]\\}$ is a Bollobás system, and

$\sum_{i=1}^{m}\left(\binom{|A_{i}\cap X_{1}|+|B_{i}\cap X_{1}|}{|A_{i}\cap
X_{1}|}\binom{|A_{i}\cap X_{2}|+|B_{i}\cap X_{2}|}{|A_{i}\cap
X_{2}|}\right)^{-1}=1+k.$

For the general case of an arbitrary $r$, taking the product of all $l\in[r]$
on inequality (2), we have

$\sum_{i=1}^{m}\left(\prod_{k=1}^{r}\binom{|A_{i}\cap X_{k}|+|B_{i}\cap
X_{k}|}{|A_{i}\cap
X_{k}|}\right)^{-1}\leqslant\left(\prod_{k=1}^{r}(1+n_{k})\right)^{\frac{r-1}{r}}\leqslant\left(1+\frac{n}{r}\right)^{r-1},$

which is the statement of Theorem 1.9.

A family $\mathcal{F}$ of subsets is called an anti-chain, if $A\not\subseteq
B$ for any $A,B\in\mathcal{F}$. The following LYM-inequality on anti-chains is
introduced by Yamamoto [20], Meshalkin [15], and Lubell [14], and plays an
important role in extremal set theory.

######  Theorem 3.2 (LYM-inequality [14, 15, 20]).

Suppose $F_{1},\ldots,F_{m}\subseteq[n]$, and the family
$\mathcal{F}=\\{F_{1},\ldots,F_{m}\\}$ is an anti-chain. Then we have

$\sum_{i=1}^{m}\binom{n}{|F_{i}|}^{-1}\leqslant 1.$

Let $A_{i}=F_{i},B_{i}=[n]\setminus F_{i}$, and
$\mathcal{P}=\\{(A_{i},B_{i})\mid i\in[m]\\}$. Then $\mathcal{F}$ is an anti-
chain implies $\mathcal{P}$ is a Bollobás system. So Theorem 1.8 and Theorem
1.9 lead to the following two variations of LYM-inequality.

######  Corollary 3.3.

Suppose $X=[n]$ is the disjoint union of $X_{1}$ and $X_{2}$, and for some
$F_{1},\ldots,F_{m}\subseteq[n]$, the family
$\mathcal{F}=\\{F_{1},\ldots,F_{m}\\}$ is an anti-chain. Then we have

$\sum_{i=1}^{m}\left(\binom{|X_{1}|}{|F_{i}\cap
X_{1}|}\binom{|X_{2}|}{|F_{i}\cap X_{2}|}\right)^{-1}\leqslant
1+\left\lfloor\frac{n}{2}\right\rfloor.$

######  Corollary 3.4.

Suppose $X=[n]$ is the disjoint union of $X_{1},\ldots,X_{r}$, and for some
$F_{1},\ldots,F_{m}\subseteq[n]$, the family
$\mathcal{F}=\\{F_{1},\ldots,F_{m}\\}$ is an anti-chain. Then we have

$\sum_{i=1}^{m}\left(\prod_{k=1}^{r}\binom{|X_{k}|}{|F_{i}\cap
X_{k}|}\right)^{-1}\leqslant\left(1+\frac{n}{r}\right)^{r-1}.$

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