Ramsey numbers for trees

Authors

  • Zhi-Hong Sun

Keywords:

Ramsey number, tree, Tur\'an's problem

Abstract

For $n\ge 4$ let $T_n^*=(V,E)$ be the tree on $n$ vertices with $V=\{v_0,v_1,\ldots,$ $v_{n-1}\}$ and $E=\{v_0v_1,\ldots,v_0v_{n-3},v_{n-3}v_{n-2},v_{n-2}v_{n-1}\}$. In the paper we evaluate the Ramsey number $r(T_m,T_n^*)$ for $T_m\in\{P_m,K_{1,m-1},T_m^*\}$. As examples, for $n\ge 8$ we have $r(P_n,T_n^*)=r(T_n^*,T_n^*)=2n-5$, for $n>m\ge 6$, $m-1\nmid (n-3)$ and $mn\equiv 0\pmod 2$ we have $r(K_{1,m-1},T_n^*)=n+m-4$, for $m\ge 6$ and $n\ge (m-3)^2+2$ we have $r(T_m,T_n^*)=n+m-3$ or $n+m-4$ according as $m-1\mid (n-3)$ or $m-1\nmid (n-3)$. DOI: 10.1017/S0004972711003388

Published

2012-06-25

Issue

Vol. 86 No. 1 (2012)

Section

Articles
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