论文标题
拉姆西 - 图兰理论中的Andrásfai和Vega图
Andrásfai and Vega graphs in Ramsey-Turán theory
论文作者
论文摘要
给定的正整数$ n \ ge s $，我们让$ {\ mathrm {ex}}（n，s）$表示$ n $ dertices上的最大边数$ g $，带有$α（g）\ le s $。在六十年代初期，安德拉斯法（Andrásfai）猜想，对于$ n/3 <s <n/2 $，函数$ {\ mathrm {ex}}}}（n，s）$是分段四边形的，临界值为$ s/n = {k}/（k}/（k}/（{k}/（{3k-1}}））$。我们确认，每当$ s/n $比临界值稍大时，确实就是这种情况，从而确定所有$ n $和$ n $和$ s $的$ {\ mathrm {ex}}（n，s）$ $γ_k=θ（k^{ - 6}）$。
Given positive integers $n\ge s$, we let ${\mathrm{ex}}(n,s)$ denote the maximum number of edges in a triangle-free graph $G$ on $n$ vertices with $α(G)\le s$. In the early sixties Andrásfai conjectured that for $n/3<s<n/2$ the function ${\mathrm{ex}}(n, s)$ is piecewise quadratic with critical values at $s/n={k}/({3k-1})$. We confirm that this is indeed the case whenever $s/n$ is slightly larger than a critical value, thus determining ${\mathrm{ex}}(n,s)$ for all $n$ and $s$ such that $s/n\in [{k}/({3k-1}), {k}/({3k-1})+γ_k]$, where $γ_k=Θ(k^{-6})$.
