# Three-Color Ramsey Numbers For Paths

@article{Gyrfs2007ThreeColorRN,
title={Three-Color Ramsey Numbers For Paths},
author={Andr{\'a}s Gy{\'a}rf{\'a}s and Mikl{\'o}s Ruszink{\'o} and G{\'a}bor N. S{\'a}rk{\"o}zy and Endre Szemer{\'e}di},
journal={Combinatorica},
year={2007},
volume={27},
pages={35-69}
}
• Published 2007
• Mathematics
• Combinatorica
We prove—for sufficiently large n—the following conjecture of Faudree and Schelp: $$R{\left( {P_{n} ,P_{n} ,P_{n} } \right)} = \left\{ {\begin{array}{*{20}c} {{2n - 1{\kern 1pt} \;{\text{for}}\;{\text{odd}}\;n,}} \\ {{{\text{2n - 2}}\;{\text{for}}\;{\text{even}}\;n,}} \\ \end{array} } \right.$$ , for the three-color Ramsey numbers of paths on n vertices.
73 Citations
Diagonal Ramsey Numbers of Loose Cycles in Uniform Hypergraphs
• Mathematics
SIAM J. Discret. Math.
• 2017
This paper investigates to determining the exact value of diagonal Ramsey number of $\mathcal{C}^k_n$ and shows that for $n\geq 2$ and $k \geq 8$ the Ramsey number is R(k-1)n+\lfloor\frac{n-1}{2}\rfloor.
On Some Three Color Ramsey Numbers for Paths, Cycles, Stripes and Stars
• Mathematics
Graphs Comb.
• 2019
The main result of the paper is a theorem which establishes the connection between the multicolor Ramsey number and the appropriateMulticolor bipartite Ramsey number together with the ordinary Ramsey number.
On some Multicolor Ramsey Properties of Random Graphs
• Mathematics
SIAM J. Discret. Math.
• 2017
It is shown that $5n/2-15/2 \le \hat{R}(P_n) \le 74n$ for $n$ sufficiently large, which improves the previous lower bound and improves the upper bound.
Monochromatic loose paths in multicolored k-uniform cliques
• Mathematics
Discret. Math. Theor. Comput. Sci.
• 2019
There is an algorithm such that for every $r$-edge-coloring of the edges of the complete $k$-uniform hypergraph, it finds a monochromatic copy of P_\ell^{(k)}$in time at most$cn^k\$.
Bipartite Ramsey Numbers of Cycles for Random Graphs
• Mathematics
Graphs Comb.
• 2021
For graphs G and H, G → k H signify that any k -edge coloring of G contains a monochromatic H as a subgraph.
Monochromatic Matchings in the Shadow Graph of Almost Complete Hypergraphs
• Mathematics
• 2010
Edge colorings of r-uniform hypergraphs naturally define a multicoloring on the 2-shadow, i.e., on the pairs that are covered by hyperedges. We show that in any (r – 1)-coloring of the edges of an
Coverings by Few Monochromatic Pieces: A Transition Between Two Ramsey Problems
• Mathematics
Graphs Comb.
• 2015
A problem that connects areas of Ramsey theory is proposed: for a fixed positive integers s ≤ t, at least how many vertices can be covered by the vertices of no more than s monochromatic members of F in every edge coloring of Kn with t colors.
On the Multi-Colored Ramsey Numbers of Paths and Even Cycles
The upper bound on the multi-color Ramsey numbers of paths and even cycles is improved and a stability version of the Erdős-Gallai theorem is introduced that may be of independent interest.