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Paper   IPM / M / 17596
School of Mathematics
  Title:   On Ramsey numbers of 3-uniform Berge cycles
1.  Leila Maherani
2.  Maryam Shahsiah
  Status:   Published
  Journal: Discrete Math.
  Vol.:  347
  Year:  2024
  Pages:   113877
  Supported by:  IPM
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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