Ramsey numbers for trees

TL;DR

This paper calculates specific Ramsey numbers involving particular trees and connected graphs, providing exact values and conditions for various cases, advancing understanding of tree-related Ramsey theory.

Contribution

The paper determines exact Ramsey numbers for certain trees and graphs, introducing new results for trees with maximum degree n-2 and specific configurations.

Findings

For n≥8, r(T_n',T_n^*)=r(T_n^*,T_n^*)=2n-5

For n>m≥7, r(K_{1,m-1},T_n^*)=m+n-3 or m+n-4 depending on divisibility

For m≥7, n≥(m-3)^2+2, r(T_m^*,T_n^*)=m+n-3 or m+n-4 depending on divisibility

Abstract

For $n\ge 5$ let $T_n'$ denote the unique tree on $n$ vertices with $\Delta(T_n')=n-2$, and 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 this paper we evaluate the Ramsey numbers $r(G_m,T_n')$ and $r(G_m,T_n^*)$, where $G_m$ is a connected graph of order $m$. As examples, for $n\ge 8$ we have $r(T_n',T_n^*)=r(T_n^*,T_n^*)=2n-5$, for $n>m\ge 7$ we have $r(K_{1,m-1},T_n^*)=m+n-3$ or $m+n-4$ according as $m-1\mid (n-3)$ or $m-1\nmid (n-3)$, for $m\ge 7$ and $n\ge (m-3)^2+2$ we have $r(T_m^*,T_n^*)=m+n-3$ or $m+n-4$ according as $m-1\mid (n-3)$ or $m-1\nmid (n-3)$.