SIAM Journal on Discrete Mathematics, Volume 34, Issue 4, Page 2556-2582, January 2020.
How many cliques can a graph on $n$ vertices have with a forbidden substructure? Extremal problems of this sort have been studied for a long time. This paper studies the maximum possible number of cliques in a graph on $n$ vertices with a forbidden clique subdivision or immersion. We prove for $t$ sufficiently large that every graph on $n \geq t$ vertices with no $K_t$-immersion has at most $n2^{t+\log_2^2 t}$ cliques, which is sharp apart from the $2^{O(\log^2 t)}$ factor. We also prove that the maximum number of cliques in an $n$-vertex graph with no $K_t$-subdivision is at most $2^{1.817t }n$ for sufficiently large $t$. This improves on the best known exponential constant by Lee and Oum. We conjecture that the optimal bound is $3^{2t/3 +o(t)}n$, as we proved for minors in place of subdivision in earlier work.

中文翻译：

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关于禁止细分或浸入的图形中的集团数量

SIAM离散数学杂志，第34卷，第4期，第2556-2582页，2020年1月。
在$ n $个顶点上的图具有禁止子结构的几个小组？这种极端问题已被研究了很长时间。本文研究了在具有禁止的组细分或浸入的$ n $个顶点上的图形中的组的最大可能数目。对于$ t $，我们证明足够大，以至于没有$ K_t $浸入的$ n \ geq t $顶点上的每个图最多具有$ n2 ^ {t + \ log_2 ^ 2 t} $组，这与$ 2截然不同^ {O（\ log ^ 2 t）} $因子。我们还证明了，对于没有足够的$ K_t $细分的$ n $顶点图中的最大派系数量最多为$ 2 ^ {1.817t} n $。这是Lee和Oum最著名的指数常数的改进。我们推测最佳边界是$ 3 ^ {2t / 3 + o（t）} n $，正如我们在早期工作中用未成年人代替细分所证明的那样。