Ramsey numbers for trees II

Sun, Zhi-Hong. "Ramsey numbers for trees II." Czechoslovak Mathematical Journal 71.2 (2021): 351-372. <http://eudml.org/doc/297748>.

@article{Sun2021,
abstract = {Let $r(G_1, G_2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. For $n_1\ge n_2\ge 1$ let $S(n_1,n_2)$ be the double star given by $V(S(n_1,n_2))=\lbrace v_0,v_1,\ldots ,v_\{n_1\},w_0$, $w_1,\ldots ,w_\{n_2\}\rbrace $ and $E(S(n_1,n_2))=\lbrace v_0v_1,\ldots ,v_0v_\{n_1\},v_0w_0, w_0w_1,\ldots ,w_0w_\{n_2\}\rbrace $. We determine $r(K_\{1,m-1\},$$S(n_1,n_2))$ under certain conditions. For $n\ge 6$ let $T_n^3=S(n-5,3)$, $T_n^\{\prime \prime \}=(V,E_2)$ and $T_n^\{\prime \prime \prime \} =(V,E_3)$, where $V=\lbrace v_0,v_1,\ldots ,v_\{n-1\}\rbrace $, $E_2=\lbrace v_0v_1,\ldots ,v_0v_\{n-4\},v_1v_\{n-3\}$, $v_1v_\{n-2\}, v_2v_\{n-1\}\rbrace $ and $E_3=\lbrace v_0v_1,\ldots , v_0v_\{n-4\},v_1v_\{n-3\},$$v_2v_\{n-2\},v_3v_\{n-1\}\rbrace $. We also obtain explicit formulas for $r(K_\{1,m-1\},T_n)$, $r(T_m^\{\prime \},T_n)$$(n\ge m+3)$, $r(T_n,T_n)$, $r(T_n^\{\prime \},T_n)$ and $r(P_n,T_n)$, where $T_n\in \lbrace T_n^\{\prime \prime \},T_n^\{\prime \prime \prime \},T_n^3\rbrace $, $P_n$ is the path on $n$ vertices and $T_n^\{\prime \}$ is the unique tree with $n$ vertices and maximal degree $n-2$.},
author = {Sun, Zhi-Hong},
journal = {Czechoslovak Mathematical Journal},
keywords = {Ramsey number; tree; Turán's problem},
language = {eng},
number = {2},
pages = {351-372},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Ramsey numbers for trees II},
url = {http://eudml.org/doc/297748},
volume = {71},
year = {2021},
}

TY - JOUR
AU - Sun, Zhi-Hong
TI - Ramsey numbers for trees II
JO - Czechoslovak Mathematical Journal
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 71
IS - 2
SP - 351
EP - 372
AB - Let $r(G_1, G_2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. For $n_1\ge n_2\ge 1$ let $S(n_1,n_2)$ be the double star given by $V(S(n_1,n_2))=\lbrace v_0,v_1,\ldots ,v_{n_1},w_0$, $w_1,\ldots ,w_{n_2}\rbrace $ and $E(S(n_1,n_2))=\lbrace v_0v_1,\ldots ,v_0v_{n_1},v_0w_0, w_0w_1,\ldots ,w_0w_{n_2}\rbrace $. We determine $r(K_{1,m-1},$$S(n_1,n_2))$ under certain conditions. For $n\ge 6$ let $T_n^3=S(n-5,3)$, $T_n^{\prime \prime }=(V,E_2)$ and $T_n^{\prime \prime \prime } =(V,E_3)$, where $V=\lbrace v_0,v_1,\ldots ,v_{n-1}\rbrace $, $E_2=\lbrace v_0v_1,\ldots ,v_0v_{n-4},v_1v_{n-3}$, $v_1v_{n-2}, v_2v_{n-1}\rbrace $ and $E_3=\lbrace v_0v_1,\ldots , v_0v_{n-4},v_1v_{n-3},$$v_2v_{n-2},v_3v_{n-1}\rbrace $. We also obtain explicit formulas for $r(K_{1,m-1},T_n)$, $r(T_m^{\prime },T_n)$$(n\ge m+3)$, $r(T_n,T_n)$, $r(T_n^{\prime },T_n)$ and $r(P_n,T_n)$, where $T_n\in \lbrace T_n^{\prime \prime },T_n^{\prime \prime \prime },T_n^3\rbrace $, $P_n$ is the path on $n$ vertices and $T_n^{\prime }$ is the unique tree with $n$ vertices and maximal degree $n-2$.
LA - eng
KW - Ramsey number; tree; Turán's problem
UR - http://eudml.org/doc/297748
ER -