Turán's problem and Ramsey numbers for trees

Details Turán's problem and Ramsey numbers for trees Turán's problem and Ramsey numbers for trees

2011-10-12

arxiv.org/abs/1110.2725v2

Mathematics

Combinatorics

23 pages

Scientific paper

Let $T_n^1=(V,E_1)$ and $T_n^2=(V,E_2)$ be the trees on $n$ vertices with $V=\{v_0,v_1,...,v_{n-1}\}$, $E_1=\{v_0v_1,...,v_0v_{n-3},v_{n-4}v_{n-2},v_{n-3}v_{n-1}\}$ and $E_2=\{v_0v_1,...,v_0v_{n-3},v_{n-3}v_{n-2}, v_{n-3}v_{n-1}\}$. In the paper, for $p\ge n\ge 8$ we obtain explicit formulas for $ex(p;T_n^1)$ and $ex(p;T_n^2)$, where $ex(p;L)$ denote the maximal number of edges in a graph of order $p$ not containing any graphs in $L$. Let $r(G\sb 1, G\sb 2)$ be the Ramsey number of the two graphs $G_1$ and $G_2$. In the 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 $\Delta(T_m)\le m-3$.

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