A note on the size Ramsey numbers for matchings versus cycles

Baskoro, Edy Tri, and Vetrík, Tomáš. "A note on the size Ramsey numbers for matchings versus cycles." Mathematica Bohemica 146.2 (2021): 229-234. <http://eudml.org/doc/297486>.

@article{Baskoro2021,
abstract = {For graphs $G$, $F_1$, $F_2$, we write $G \rightarrow (F_1, F_2)$ if for every red-blue colouring of the edge set of $G$ we have a red copy of $F_1$ or a blue copy of $F_2$ in $G$. The size Ramsey number $\hat\{r\}(F_1, F_2)$ is the minimum number of edges of a graph $G$ such that $G \rightarrow (F_1, F_2)$. Erdős and Faudree proved that for the cycle $C_n$ of length $n$ and for $t \ge 2$ matchings $tK_2$, the size Ramsey number $\hat\{r\} (tK_2, C_n) < n + (4t+3) \sqrt\{n\}$. We improve their upper bound for $t = 2$ and $t=3$ by showing that $\hat\{r\} (2K_2, C_n) \le n + 2 \sqrt\{3n\} + 9$ for $n \ge 12$ and $\hat\{r\} (3K_2, C_n) < n + 6 \sqrt\{n\} + 9$ for $n \ge 25$.},
author = {Baskoro, Edy Tri, Vetrík, Tomáš},
journal = {Mathematica Bohemica},
keywords = {size Ramsey number; matching; cycle},
language = {eng},
number = {2},
pages = {229-234},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on the size Ramsey numbers for matchings versus cycles},
url = {http://eudml.org/doc/297486},
volume = {146},
year = {2021},
}

TY - JOUR
AU - Baskoro, Edy Tri
AU - Vetrík, Tomáš
TI - A note on the size Ramsey numbers for matchings versus cycles
JO - Mathematica Bohemica
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 146
IS - 2
SP - 229
EP - 234
AB - For graphs $G$, $F_1$, $F_2$, we write $G \rightarrow (F_1, F_2)$ if for every red-blue colouring of the edge set of $G$ we have a red copy of $F_1$ or a blue copy of $F_2$ in $G$. The size Ramsey number $\hat{r}(F_1, F_2)$ is the minimum number of edges of a graph $G$ such that $G \rightarrow (F_1, F_2)$. Erdős and Faudree proved that for the cycle $C_n$ of length $n$ and for $t \ge 2$ matchings $tK_2$, the size Ramsey number $\hat{r} (tK_2, C_n) < n + (4t+3) \sqrt{n}$. We improve their upper bound for $t = 2$ and $t=3$ by showing that $\hat{r} (2K_2, C_n) \le n + 2 \sqrt{3n} + 9$ for $n \ge 12$ and $\hat{r} (3K_2, C_n) < n + 6 \sqrt{n} + 9$ for $n \ge 25$.
LA - eng
KW - size Ramsey number; matching; cycle
UR - http://eudml.org/doc/297486
ER -