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| Paper IPM / M / 17596 |
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| Abstract: | |||||
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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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