# On some Multicolor Ramsey Properties of Random Graphs

@article{Dudek2017OnSM,
title={On some Multicolor Ramsey Properties of Random Graphs},
author={Andrzej Dudek and Paweł Prałat},
journal={SIAM J. Discret. Math.},
year={2017},
volume={31},
pages={2079-2092}
}
• Published 11 January 2016
• Mathematics
• SIAM J. Discret. Math.
The size-Ramsey number $\hat{R}(F)$ of a graph $F$ is the smallest integer $m$ such that there exists a graph $G$ on $m$ edges with the property that any colouring of the edges of $G$ with two colours yields a monochromatic copy of $F$. In this paper, first we focus on the size-Ramsey number of a path $P_n$ on $n$ vertices. In particular, we show that $5n/2-15/2 \le \hat{R}(P_n) \le 74n$ for $n$ sufficiently large. (The upper bound uses expansion properties of random $d$-regular graphs.) This…
Size Ramsey numbers of paths
• Mathematics
• 2018
The size Ramsey number $\widehat{R}({F},r)$ is the minimum integer $m$ such that there exists a graph $G$ on $m$ edges such that every coloring of the edges of $G$ with $r$ colors yields a
Note on the Multicolour Size-Ramsey Number for Paths,
• Mathematics
Electron. J. Comb.
• 2018
This short note gives an alternative proof of the recent result of Krivelevich that $\hat{R}(P_n,r) = O((\log r)r^2 n)$.
Size-Ramsey numbers of cycles versus a path
• Mathematics
Discret. Math.
• 2018
On the Size-Ramsey Number of Tight Paths
• Mathematics, Computer Science
SIAM J. Discret. Math.
• 2018
The size-Ramsey number of tight paths is improved by showing that $\hat R(\mathcal{P}_{n,k-1}^{(k)}, r) = O(r^k (n\log n)^{k/2})$ for all $k\geq 3$ and $r \geq 2$.
The multicolour size-Ramsey number of powers of paths
• Mathematics
J. Comb. Theory, Ser. B
• 2020
Ordered size Ramsey number of paths
• Mathematics
Discret. Appl. Math.
• 2020
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.
On the Size-Ramsey Number of Cycles
• Mathematics
Combinatorics, Probability and Computing
• 2019
Various upper bounds for the size-Ramsey numbers of cycles are given, including an alternative proof of ${\hat R_k}({C_n}) \le {c-k}n$ , avoiding use of the regularity lemma.
Cycle Ramsey numbers for random graphs
• Mathematics
• 2018
Let $C_{n}$ be a cycle of length $n$. As an application of Szemeredi's regularity lemma, Łuczak ($R(C_n,C_n,C_n)\leq (4+o(1))n$, J. Combin. Theory Ser. B, 75 (1999), 174--187) in fact established