Geodesic
On the multi-colored Ramsey numbers of paths and even cycles
Gábor N. Sárközy1
1Worcester Polytechnic Institute
The electronic journal of combinatorics, Tome 23 (2016) no. 3
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In this paper we improve the upper bound on the multi-color Ramsey numbers of paths and even cycles. More precisely, we prove the following. For every $r\geq 2$ there exists an $n_0=n_0(r)$ such that for $n\geq n_0$ we have $$R_r(P_n) \leq \left( r - \frac{r}{16r^3+1} \right) n.$$ For every $r\geq 2$ and even $n$ we have $$R_r(C_n) \leq \left( r - \frac{r}{16r^3+1} \right) n + o(n) \text{ as }n\rightarrow \infty.$$ The main tool is a stability version of the Erdős-Gallai theorem that may be of independent interest.
DOI : 10.37236/5663
Classification : 05C55, 05D10
Mots-clés : Ramsey numbers, paths, cycles

Gábor N. Sárközy  1

1 Worcester Polytechnic Institute 
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     author = {G\'abor N. S\'ark\"ozy},
     title = {On the multi-colored {Ramsey} numbers of paths and even cycles},
     journal = {The electronic journal of combinatorics},
     year = {2016},
     volume = {23},
     number = {3},
     doi = {10.37236/5663},
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     url = {http://geodesic.mathdoc.fr/articles/10.37236/5663/}
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Gábor N. Sárközy. On the multi-colored Ramsey numbers of paths and even cycles. The electronic journal of combinatorics, Tome 23 (2016) no. 3. doi: 10.37236/5663

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