On the Multi-Colored Ramsey Numbers of Paths and Even Cycles
@article{Srkzy2016OnTM, title={On the Multi-Colored Ramsey Numbers of Paths and Even Cycles}, author={G{\'a}bor N. S{\'a}rk{\"o}zy}, journal={Electron. J. Comb.}, year={2016}, volume={23}, pages={3} }
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…
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References
SHOWING 1-10 OF 24 REFERENCES
On Three-Color Ramsey Number of Paths
- MathematicsGraphs Comb.
- 2015
This paper shows that if (n,m)≠(3,3), (3,4) and m≥n, then R(P_3,P_n, P_m) = 2m+n-1, and R (P3,mK2,nK2) =2m-1 for $$m\ge n\ge 3$$m≥ n≥3.
ON MAXIMAL PATHS AND CIRCUITS OF GRAPHS
- Mathematics
- 2002
In 1940 TURIN raised the following question: if the number of nodes, n, of a graph’ is prescribed and if I is an integer cs n, what is the number of edges which the graph has to contain in order to…
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…
Small Ramsey Numbers
- Mathematics, Computer Science
- 2011
We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are…
Regular Partitions of Graphs
- Mathematics
- 1975
Abstract : A crucial lemma in recent work of the author (showing that k-term arithmetic progression-free sets of integers must have density zero) stated (approximately) that any large bipartite graph…
R(Cn, Cn, Cn)<=(4+o(1)) n
- MathematicsJ. Comb. Theory, Ser. B
- 1999
It is shown that the value of R(Cn, Cn , Cn ; Cn) does not grow with n much faster than 4n, and this problem is not able to settle.