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Paper IPM / M / 17596

School of Mathematics

Title:

On Ramsey numbers of 3-uniform Berge cycles

Author(s):

1.	Leila Maherani
2.	Maryam Shahsiah

Status:

Published

Journal:

Discrete Math.

Vol.:

347

Year:

2024

Pages:

113877

Supported by:

IPM

Abstract:

For an arbitrary graph

$G$ , a hypergraph

$\mathcal{H}$ is called Berge-

$G$ if there is an injection

$i: V(G)\longrightarrow V(\mathcal{H})$ and a bijection

$\psi :E(G)\longrightarrow E(\mathcal{H})$ such that for each

$e=uv\in E(G)$ , we have

$\{i(u),i(v)\}\subseteq \psi (e)$ . We denote by

$\mathcal{B}^rG$ , the family of

$r$ -uniform Berge-

$G$ hypergraphs. For families

$\mathcal{F}_1, \mathcal{F}_2,\ldots, \mathcal{F}_t$ of

$r$ -uniform hypergraphs, the Ramsey number

$R(\mathcal{F}_1, \mathcal{F}_2,\ldots, \mathcal{F}_t)$ is the minimum integer

$n$ such that in every hyperedge coloring of the complete

$r$ -uniform hypergraph on

$n$ vertices with

$t$ colors, there exists

$i$ ,

$1\leq i\leq t$ , such that there is a monochromatic copy of a hypergraph in

$\mathcal{F}_i$ of color

$i$ . Recently, the extremal problems of Berge hypergraphs have received considerable attention.\\ In this paper, we focus on Ramsey numbers involving

$3$ -uniform Berge cycles and prove that for

$n \geq 4$ ,

$R(\mathcal{B}^3C_n,\mathcal{B}^3C_n,\mathcal{B}^3C_3)=n+1.$ Moreover, for

$m\geq 11$ and

$m \geq n\geq 5$ , we show that

$R(\mathcal{B}^3K_m,\mathcal{B}^3C_n)= m+\lfloor \frac{n-1}{2}\rfloor -1.$ This is the first result of Ramsey number for two different families of Berge hypergraphs.

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“School of Mathematics”

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