One of the major problems in combinatorics is to determine the number of r-uniform hypergraphs (r-graphs) on n vertices which are free of certain forbidden structures. This problem dates back to the work of Erdős, Kleitman and Rothschild, who showed that the number of $K_r$-free graphs on n vertices is $2^{\mathrm{\text{ex}}(n,K_r)+o(n^2)}$. Their work was later extended to forbidding graphs as induced subgraphs by Prömel and Steger.

Here, we consider one of the most basic counting problems for 3-graphs. Let $E_1$ be the 3-graph with 4 vertices and 1 edge. What is the number of induced $\{K_4^3,E_1\}$-free 3-graphs on n vertices? We show that the number of such 3-graphs is of order $n^{\mathrm{\Theta }(n^2)}$. More generally, we determine asymptotically the number of induced $\mathcal{F}$-free 3-graphs on n vertices for all families $\mathcal{F}$ of 3-graphs on 4 vertices. We also provide upper bounds on the number of r-graphs on n vertices which do not induce $i\in L$ edges on any set of k vertices, where $L\subseteq \{0,1,\dots ,\left(\begin{array}{c}k\\ r\end{array}\right)\}$ is a list which does not contain 3 consecutive integers in its complement. Our bounds are best possible up to a constant multiplicative factor in the exponent when $k=r+1$. The main tool behind our proof is counting the solutions of a constraint satisfaction problem.

组合数学中的主要问题之一是确定没有某些禁止结构的n个顶点上的r一致超图（r图）的数量。这个问题可以追溯到 Erdős、Kleitman 和 Rothschild 的工作，他们表明${ķ}_r$-n个顶点上的自由图是$2^{\mathrm{\text{前任}}(n,{ķ}_r)+○(n^2)}$. 他们的工作后来被 Prömel 和 Steger 扩展到禁止图作为诱导子图。

在这里，我们考虑 3-graph 最基本的计数问题之一。让${乙}_1$是具有 4 个顶点和 1 条边的 3-图。诱导次数是多少$\{{ķ}_4^3,{乙}_1\}$-n个顶点上的自由 3-图？我们证明了这样的 3-图的数量是有序的$n^{\mathrm{\theta }(n^2)}$. 更一般地，我们渐近地确定诱导的数量$\mathcal{F}$- 所有族的n个顶点上的免费 3 图$\mathcal{F}$4 个顶点上的 3 个图。我们还提供了不诱导的n个顶点上的r图数量的上限$\mathrm{一世}\in \mathrm{大号}$任意k个顶点集合上的边，其中$\mathrm{大号}\subseteq \{0,1,\dots ,\left(\begin{array}{c}ķ\\ r\end{array}\right)\}$是一个在其补码中不包含 3 个连续整数的列表。我们的界限是最好的，直到指数中的一个常数乘法因子，当$ķ=r+1$. 我们证明背后的主要工具是计算约束满足问题的解决方案。