“School of Mathematics”
Back to Papers HomeBack to Papers of School of Mathematics
| Paper IPM / M / 18009 |
|
| Abstract: | |
|
For given $k$-uniform hypergraphs $\mathcal{G}$ and $\mathcal{H}$, the Ramsey number $R(\mathcal{G},\mathcal{H})$ is the smallest positive integer $N$ such that in every red-blue coloring of the edges of the complete $k$-uniform hypergraph on $n$ vertices there is either a red copy of $\mathcal{G}$ or a blue copy of $\mathcal{H}$. In this paper, results are given which permit the $R(m\mathcal{G},n\mathcal{H})$ to be evaluated exactly when $m$ or $n$ is large and $\mathcal{G}$ is a
$k$-uniform hypergraph with the maximum independent set that intersects each edge in $k-1$ vertices and $\mathcal{H}$ is a $k$-uniform hypergraph with a vertex so that the hypergraph induced by the edges containing this vertex is a star. There are several examples for such $\mathcal{G}$ and $\mathcal{H}$, among
them are any disjoint union of
$k$-uniform hypergraphs involving loose paths, loose cycles,
tight paths, tight cycles, stars, Kneser hypergraphs and complete $k$-uniform $k$-partite hypergraphs for $\mathcal{G}$ and linear hypergraphs for $\mathcal{H}$. As an application, $R(m\mathcal{G},n\mathcal{H})$ is determined when $m$ or $n$ is large and $\mathcal{G}$, $\mathcal{H}$ are either loose paths, loose cycles, tight paths or stars. Moreover, for given $k$-uniform hypergraphs $\mathcal{G}$ and $\mathcal{H}$ and positive integers $m, n$, some bounds are given for $R(m\mathcal{G},n\mathcal{H})$ which enable us to compute $R(m\mathcal{G},n\mathcal{H})$ when $m\geq n\geq 1$ and
$\mathcal{G}, \mathcal{H}$ are either 3-uniform loose path $\mathcal{P}_r^3$ or loose cycle $\mathcal{C}_r^3$: We shall show
that for every $m\geq n\geq 1$ and $r\geq s$, $R(m\mathcal{C}_r^3,n\mathcal{C}_s^3)=2rm+\Big\lfloor\frac{s+1}{2}\Big\rfloor n-1,$ and $R(m\mathcal{P}_r^3,n\mathcal{P}_s^3)=(2r+1)m+\Big\lfloor\frac{s+1}{2}\Big\rfloor n-1.$
Download TeX format |
|
| back to top | |
