论文标题
关于Erdős-Rademacher问题的稳定性
On stability of the Erdős-Rademacher Problem
论文作者
论文摘要
Mantel的定理指出，每个$ n $ vertex图形都带有$ \ lfloor \ frac {n^2} {4} {4} \ rfloor +t $ edge，其中$ t> 0 $包含一个三角形。确定该图中最小三角形数量的问题通常称为ERDőS-RADEMACHER问题。 Lovász和Simonovits证明，每个图中的每个图中至少都有$ t \ lfloor n/2 \ rfloor $三角形。 Katona和Xiao在其他条件下认为没有$ s-1 $的顶点涵盖所有三角形的情况下，也考虑了相同的问题。他们解决了案例$ t = 1 $，$ s = 2 $。解决他们的猜想，我们确定每对$ s $和$ t $的最小三角形数量，当$ n $足够大。此外，我们解决了Katona和Xiao的另一个猜想，我们扩展了考虑集团而不是三角形的理论。
Mantel's theorem states that every $n$-vertex graph with $\lfloor \frac{n^2}{4} \rfloor +t$ edges, where $t>0$, contains a triangle. The problem of determining the minimum number of triangles in such a graph is usually referred to as the Erdős-Rademacher problem. Lovász and Simonovits proved that there are at least $t\lfloor n/2 \rfloor$ triangles in each of those graphs. Katona and Xiao considered the same problem under the additional condition that there are no $s-1$ vertices covering all triangles. They settled the case $t=1$ and $s=2$. Solving their conjecture, we determine the minimum number of triangles for every fixed pair of $s$ and $t$, when $n$ is sufficiently large. Additionally, solving another conjecture of Katona and Xiao, we extend the theory for considering cliques instead of triangles.
