#
The Size, Multipartite Ramsey Numbers for nK_{2} Versus Path–Path and Cycle

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Theorem**

**2.**

## 2. Proof of Theorem 1

**Lemma**

**1.**

**Proof.**

**Observation**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

**Claim**

**1.**

**Claim**

**2.**

**Claim**

**3.**

**Claim**

**4.**

**Lemma**

**6.**

**Lemma**

**7.**

**Proof.**

**Claim**

**5.**

**Claim**

**6.**

**Claim**

**7.**

## 3. Proof of Theorem 2

**Proposition**

**1.**

**Proof.**

**Lemma**

**8.**

**Proof.**

**Lemma**

**9.**

**Proof.**

**Lemma**

**10.**

**Proof.**

**Lemma**

**11.**

**Proof.**

**Claim**

**8.**

**Claim**

**9.**

## 4. Concluding Remarks and Further Works

- ${m}_{j}({K}_{1,2},{P}_{4},n{K}_{2})\ge \lfloor \frac{2n}{j}\rfloor +1$ where $j,n\ge 2$;
- ${m}_{2}({K}_{1,2},{P}_{4},n{K}_{2})=n+1$ for $n\ge 2$;
- ${m}_{3}({K}_{1,2},{P}_{4},n{K}_{2})=\lfloor \frac{2n}{3}\rfloor +1$ for $n\ge 2$;
- Let $j\ge 4$ and $n\ge 2$. Given that ${m}_{j}({K}_{1,2},{P}_{4},(n-1){K}_{2})=\lfloor \frac{2(n-1)}{j}\rfloor +1$, it follows that ${m}_{j}({K}_{1,2},{P}_{4},n{K}_{2})\le \lfloor \frac{2n}{j}\rfloor +1.$

- ${m}_{3}(n{K}_{2},{C}_{7})=3$ where $n=2,3$;
- For each $n\ge 3$ we have ${m}_{3}(n{K}_{2},{C}_{7})=n$;
- For $n\ge 4$ we have ${m}_{4}(n{K}_{2},{C}_{7})=\lceil \frac{n+1}{2}\rceil $; We estimated our result for ${m}_{j}(n{K}_{2},{C}_{7})$ which holds for every $j\ge 2$, so it could be a good problem to work on.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Rowshan, Y.; Gholami, M.; Shateyi, S.
The Size, Multipartite Ramsey Numbers for *nK*_{2} Versus Path–Path and Cycle. *Mathematics* **2021**, *9*, 764.
https://doi.org/10.3390/math9070764

**AMA Style**

Rowshan Y, Gholami M, Shateyi S.
The Size, Multipartite Ramsey Numbers for *nK*_{2} Versus Path–Path and Cycle. *Mathematics*. 2021; 9(7):764.
https://doi.org/10.3390/math9070764

**Chicago/Turabian Style**

Rowshan, Yaser, Mostafa Gholami, and Stanford Shateyi.
2021. "The Size, Multipartite Ramsey Numbers for *nK*_{2} Versus Path–Path and Cycle" *Mathematics* 9, no. 7: 764.
https://doi.org/10.3390/math9070764