The inducibility of a graph H measures the maximum number of induced copies of H a large graph G can have. Generalizing this notion, we study how many induced subgraphs of fixed order k and size ℓ a large graph G on n vertices can have. Clearly, this number is $\left( {\matrix{n \cr k}}\right)$ for every n, k and $\ell \in \left\{ {0,\left( {\matrix{k \cr 2}} \right)}\right\}$. We conjecture that for every n, k and $0 \lt \ell \lt \left( {\matrix{k \cr 2}}\right)$ this number is at most $ (1/e + {o_k}(1)) {\left( {\matrix{n \cr k}} \right)}$. If true, this would be tight for ℓ ∈ {1, k − 1}.In support of our ‘Edge-statistics Conjecture’, we prove that the corresponding density is bounded away from 1 by an absolute constant. Furthermore, for various ranges of the values of ℓ we establish stronger bounds. In particular, we prove that for ‘almost all’ pairs (k, ℓ) only a polynomially small fraction of the k-subsets of V(G) have exactly ℓ edges, and prove an upper bound of $ (1/2 + {o_k}(1)){\left( {\matrix{n \cr k}}\right)}$ for ℓ = 1.Our proof methods involve probabilistic tools, such as anti-concentration results relying on fourth moment estimates and Brun’s sieve, as well as graph-theoretic and combinatorial arguments such as Zykov’s symmetrization, Sperner’s theorem and various counting techniques.

中文翻译：

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大图上的边统计

图的可归纳性H测量诱导拷贝的最大数量H大图G可以有。推广这个概念，我们研究有多少固定顺序的诱导子图ķ和大小ℓ大图G在n顶点可以有。显然，这个数字是$\left( {\matrix{n \cr k}}\right)$对于每个n,ķ和$\ell \in \left\{ {0,\left( {\matrix{k \cr 2}} \right)}\right\}$. 我们推测对于每个n,ķ和$0 \lt \ell \lt \left( {\matrix{k \cr 2}}\right)$这个数字最多$ (1/e + {o_k}(1)) {\left( {\matrix{n \cr k}} \right)}$. 如果是真的，这对ℓ∈ {1,ķ− 1}。为了支持我们的“边缘统计猜想”，我们证明了相应的密度以绝对常数为界远离 1。此外，对于不同范围的值ℓ我们建立了更强的界限。特别是，我们证明对于“几乎所有”对 (ķ,ℓ) 只是多项式的一小部分ķ- 的子集五(G) 正好ℓ边，并证明上界$ (1/2 + {o_k}(1)){\left( {\matrix{n \cr k}}\right)}$为了ℓ= 1.我们的证明方法涉及概率工具，例如依赖于四阶矩估计和布伦筛的反集中结果，以及图论和组合论证，例如 Zykov 对称化、Sperner 定理和各种计数技术。