# The Ramsey Number of Loose Triangles and Quadrangles in Hypergraphs

@article{Gyrfs2012TheRN, title={The Ramsey Number of Loose Triangles and Quadrangles in Hypergraphs}, author={Andr{\'a}s Gy{\'a}rf{\'a}s and Ghaffar Raeisi}, journal={Electron. J. Comb.}, year={2012}, volume={19}, pages={30} }

Asymptotic values of hypergraph Ramsey numbers for loose cycles (and paths) were determined recently. Here we determine some of them exactly, for example the 2-color hypergraph Ramsey number of a $k$-uniform loose 3-cycle or 4-cycle: $R(\mathcal{C}^k_3,\mathcal{C}^k_3)=3k-2$ and $R(\mathcal{C}_4^k,\mathcal{C}_4^k)=4k-3$ (for $k\geq 3$). For more than 3-colors we could prove only that $R(\mathcal{C}^3_3,\mathcal{C}^3_3,\mathcal{C}^3_3)=8$. Nevertheless, the $r$-color Ramsey number of triangles…

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## References

SHOWING 1-10 OF 12 REFERENCES

The Ramsey Number of Diamond-Matchings and Loose Cycles in Hypergraphs

- MathematicsElectron. J. Comb.
- 2008

It is proved that for $r\ge 2$, the corresponding Ramsey number is asymptotic to ${(2r-1)n\over 2r-2}$.

Ramsey Numbers Involving Cycles

- Mathematics
- 2011

Graham, Rothschild and Spencer in their book, Ramsey Theory [GRS] and Soifer in the 2009 The Mathematical Coloring Book, Mathematics of Coloring and the Colorful Life of Its Creators [Soi] present exciting developments in the history, results and people of Ramsey theory.

The Ramsey number for stripes

- MathematicsJournal of the Australian Mathematical Society
- 1975

If G1,…,Gc are graphs without loops or multiple edges there is a smallest integer r(G1,…,Gc) such that if the edges of a complete graph Kn, with n ≧ r(G1,…,Gc), are painted arbitrarily with c colours…

On Ramsey-Type Problems

- Mathematics
- 2009

In this paper, we give a brief survey on four problems of Ramsey-type. The first and second problems are concerned about a sequence of numbers. The third one appears in discrete geometry and the…

A Homological Approach to Two Problems on Finite Sets

- Mathematics
- 1999

We propose a homological approach to two conjectures descended from the Erdös-Ko-Rado Theorem, one due to Chvátal and the other to Frankl and Füredi. We apply the method to reprove, and in one case…

Some Ramsey numbers for loose cycles and paths in 3-uniform hypergraphs, in preparation

- Some Ramsey numbers for loose cycles and paths in 3-uniform hypergraphs, in preparation

Some Ramsey numbers for loose cycles and paths in 3uniform hypergraphs , in preparation