Tur\'an's problem and Ramsey numbers for trees

TL;DR

This paper derives explicit formulas for extremal graph edges avoiding certain trees and calculates specific Ramsey numbers involving these trees, advancing understanding in extremal and Ramsey graph theory.

Contribution

The paper provides new explicit formulas for extremal numbers and Ramsey numbers for particular trees, extending known results in extremal and Ramsey graph theory.

Findings

Explicit formulas for (p;T_n^1) and (p;T_n^2) for p  n  5.

Explicit formulas for r(T_m,T_n^i) where  m, n  5 and    or trees with bounded maximum degree.

Abstract

Let $T_n^1=(V,E_1)$ and $T_n^2=(V,E_2)$ be the trees on $n$ vertices with $V=\{v_0,v_1,\ldots,v_{n-1}\}$, $E_1=\{v_0v_1,\ldots,v_0v_{n-3},v_{n-4}v_{n-2},v_{n-3}v_{n-1}\}$, and $E_2=\{v_0v_1,\ldots,$ $v_0v_{n-3},v_{n-3}v_{n-2}, v_{n-3}v_{n-1}\}$. In this paper, for $p\ge n\ge 5$ we obtain explicit formulas for $\ex(p;T_n^1)$ and $\ex(p;T_n^2)$, where $\ex(p;L)$ denotes the maximal number of edges in a graph of order $p$ not containing $L$ as a subgraph. Let $r(G\sb 1, G\sb 2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. In this paper we also obtain some explicit formulas for $r(T_m,T_n^i)$, where $i\in\{1,2\}$ and $T_m$ is a tree on $m$ vertices with $\Delta(T_m)\le m-3$.