# An Erdős-Ko-Rado Type Theorem via the Polynomial Method

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1**

**.**If $n\ge 2k$ and $\mathcal{F}$ is an intersecting k-uniform family of subsets of $\left[n\right]$, then

**Theorem**

**2**

**Theorem**

**3**

**.**If $\mathcal{F}$ is a family of subsets ${F}_{i}$ of $\left[n\right]$ with $|{F}_{i}|=k$ and $|{F}_{i}|\le n-k$ that satisfies the following two conditions, for $i\ne j$

- (a)
- $1\le |{F}_{i}\cap {F}_{j}|\le k-1$
- (b)
- $1\le |{F}_{i}\cap {F}_{j}^{c}|\le k-1$

## 2. Results

**Theorem**

**4.**

- (a)
- $1\le |{F}_{i}\cap {F}_{j}|\le k-1$
- (b)
- $1\le |{F}_{i}\cap {F}_{j}^{c}|\le k-1$

**Theorem**

**5**

**.**Let $\mathcal{F}$ be a family of subsets of $\left[n\right]$ of size at most k, $1\le k\le n-1$. Suppose that for every two subsets $A,B\in \mathcal{F}$, if $A\cap B=\varnothing $, then $\left|A\right|+\left|B\right|\le k$. Then we have

**Theorem**

**6.**

- (a)
- $1\le |{F}_{i}\cap {F}_{j}|\le k-1$
- (b)
- $1\le |{F}_{i}\cap {F}_{j}^{c}|\le k-1$
- (c)
- $1\le |{F}_{i}^{c}\cap {F}_{j}^{c}|\le k-1$

## 3. Polynomial Method

**Theorem**

**7**

**.**Let ${l}_{1},{l}_{2},\cdots ,{l}_{s}<n$ be nonnegative integers. If $\mathcal{F}$ is a k-uniform family of subsets of $\left[n\right]$ such that $|A\cap B|\in L=\{{l}_{1},{l}_{2},\cdots ,{l}_{s}\}$ holds for every pair of distinct subsets $A,B\in \mathcal{F}$, then $\left|\mathcal{F}\right|\le \left(\genfrac{}{}{0pt}{}{n}{s}\right)$ holds.

**Theorem**

**8**

**.**Let ${l}_{1},{l}_{2},\cdots ,{l}_{s}<n$ be nonnegative integers. If $\mathcal{F}$ is a family of subsets of $\left[n\right]$ such that $|A\cap B|\in L=\{{l}_{1},{l}_{2},\cdots ,{l}_{s}\}$ holds for every pair of distinct subsets $A,B\in \mathcal{F}$, then $\left|\mathcal{F}\right|\le {\sum}_{k=0}^{s}\left(\genfrac{}{}{0pt}{}{n}{k}\right)$ holds.

**Proof.**

**Theorem**

**9**

**.**If $\mathcal{F}$ is a family of subsets of $\left[n\right]$ such that $|A\cap B|\equiv l\in L$ (mod p) holds for every pair of distinct subsets $A,B\in \mathcal{F}$, then $\left|\mathcal{F}\right|\le \left(\genfrac{}{}{0pt}{}{n}{\left|L\right|}\right)$ holds.

**Theorem**

**10**

**.**If $\mathcal{F}$ is a family of subsets of $\left[n\right]$ such that $|A\cap B|\equiv l\in L$ (mod p) holds for every pair of distinct subsets $A,B\in \mathcal{F}$ and $\left|A\right|\not\equiv l$ (mod p) for every $A\in \mathcal{F}$, then $\left|\mathcal{F}\right|\le {\sum}_{i=0}^{\left|L\right|}\left(\genfrac{}{}{0pt}{}{n}{i}\right)$ holds.

**Theorem**

**11**

**.**Let $K=\{{k}_{1},{k}_{2},\cdots ,{k}_{r}\}$ and $L=\{{l}_{1},{l}_{2},\cdots ,{l}_{s}\}$ be two disjoint subsets of $\{0,1,\cdots ,p-1\}$, where p is a prime, and let $\mathcal{F}$ be a family of subsets of $\left[n\right]$ whose sizes modulo p are in the set K, and $|A\cap B|$ $\left(\mathrm{mod}\phantom{\rule{4pt}{0ex}}p\right)\in L$ holds for every distinct two subsets $A,B$ in $\mathcal{F}$, then the largest size of such a family $\mathcal{F}$ is $\left(\genfrac{}{}{0pt}{}{n}{s}\right)+\left(\genfrac{}{}{0pt}{}{n}{s-1}\right)+\cdots +\left(\genfrac{}{}{0pt}{}{n}{s-r+1}\right)$ under the conditions $r(s-r+1)\le p-1$ and $n\ge s+{max}_{1\le i\le r}{k}_{i}$.

**Theorem**

**12**

**.**Let $K=\{{k}_{1},{k}_{2},\cdots ,{k}_{r}\}$ and $L=\{{l}_{1},{l}_{2},\cdots ,{l}_{s}\}$ be two disjoint subsets of $\{0,1,\cdots ,p-1\}$, where p is a prime, and let $\mathcal{F}$ be a family of subsets of $\left[n\right]$ whose sizes modulo p are in the set K, and $|A\cap B|$ $\left(\mathrm{mod}\phantom{\rule{4pt}{0ex}}p\right)\in L$ for every distinct two subsets $A,B$ in $\mathcal{F}$, then the largest size of such a family $\mathcal{F}$ is $\left(\genfrac{}{}{0pt}{}{n}{s}\right)+\left(\genfrac{}{}{0pt}{}{n}{s-1}\right)+\cdots +\left(\genfrac{}{}{0pt}{}{n}{s-r+1}\right)$ under the only condition that $n\ge s+{max}_{1\le i\le r}{k}_{i}$.

## 4. Proof of the Main Result

**Proof**

**of**

**Theorem 6.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Hwang, K.-W.; Kim, Y.; Sheikh, N.N.
An Erdős-Ko-Rado Type Theorem via the Polynomial Method. *Symmetry* **2020**, *12*, 640.
https://doi.org/10.3390/sym12040640

**AMA Style**

Hwang K-W, Kim Y, Sheikh NN.
An Erdős-Ko-Rado Type Theorem via the Polynomial Method. *Symmetry*. 2020; 12(4):640.
https://doi.org/10.3390/sym12040640

**Chicago/Turabian Style**

Hwang, Kyung-Won, Younjin Kim, and Naeem N. Sheikh.
2020. "An Erdős-Ko-Rado Type Theorem via the Polynomial Method" *Symmetry* 12, no. 4: 640.
https://doi.org/10.3390/sym12040640