Counting r-graphs without forbidden configurations

Abstract

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.

Introduction

For an r-uniform hypergraph (r-graph) F, let $\mathrm{\text{ex}}(n,F)$ denote the maximum number of edges in an F-free r-graph on n vertices. One of the central questions in extremal combinatorics is to determine the extremal number $\mathrm{\text{ex}}(n,F)$. For $r=2$, the extremal number is well-understood for all non-bipartite graphs, see [15] and [38]. However, determining the extremal number for general r-graphs is a well-known and hard problem. The simplest and still not answered question posed by Turán asks to determine the extremal number of $K_4^3$, the complete 3-graph on 4 vertices. It is widely believed that

$$\mathrm{\text{ex}}(n,K_4^3)=(\frac{5}{9}+o(1))\left(\begin{array}{c}n\\ 3\end{array}\right).$$

In a series of papers, different $K_4^3$-free 3-graphs on n vertices and $\frac{5}{9}\left(\begin{array}{c}n\\ 3\end{array}\right)+o(n^3)$ edges were constructed by Brown [11], Kostochka [23] and Fon-der-Flaass [17] and Razborov [34]. In 2008, Frohmader [19] showed that there are $\mathrm{\Omega }(6^{n/3})$ non-isomorphic r-graphs which are conjectured to be extremal. This is believed to be one of the reasons of the difficulty of this problem. For other related papers, see [3], [30], [34].

The problem of determining the extremal number can also be extended to families of induced r-graphs. For a family of r-graphs $\mathcal{F}$, let ${\mathrm{\text{ex}}}_I(n,\mathcal{F})$ denote the maximum number of edges in an induced $\mathcal{F}$-free r-graph on n vertices. In 2010, Razborov [34] used the method of flag algebras to determine ${\mathrm{\text{ex}}}_I(n,\{K_4^3,E_1\})$, where $E_1$ denotes the 3-graph with 4 vertices and 1 edge. In his paper, he showed that

$${\mathrm{\text{ex}}}_I(n,\{K_4^3,E_1\})=(\frac{5}{9}+o(1))\left(\begin{array}{c}n\\ 3\end{array}\right).$$

Later, this result was extended by Pikhurko [30], who obtained the corresponding stability result and proved that there is only one extremal induced $\{K_4^3,E_1\}$-free 3-graph on n vertices, up to isomorphism. Sometimes referred to as Turán's construction and here denoted by $C_n$, the extremal induced $\{K_4^3,E_1\}$-free 3-graph on $[n]$ is obtained as follows. Let $V_1\cup V_2\cup V_3$ be a partition of $[n]$ with $||V_i|-|V_j||\le 1$ for all $i,j\in [3]$. An edge is placed in $C_n$ if it intersects each of the classes $V_1$, $V_2$ and $V_3$, or if for some $i\in [3]$ it contains two elements of $V_i$ and one of $V_{i+1}$, where the indices are understood modulo 3. See Fig. 1 for an illustration of $C_n$.

In this paper, we first consider the problem of counting induced $\{K_4^3,E_1\}$-free 3-graphs on n vertices, which is the counting problem related to the results of Razborov [34] and Pikhurko [30]. Recently, Balogh and Mubayi [7] observed that a standard application of the hypergraph container method [4], [35] shows that the number of induced $\{K_4^3,E_1\}$-free graphs on n vertices is $2^{O(n^{8/3})}$. From the other side, we can construct a family $Q(n)$ with $2^{\mathrm{\Omega }(n^2\log n)}$ subgraphs of $C_n$ which are induced $\{K_4^3,E_1\}$-free. A 3-graph is in $Q(n)$ if it is obtained from a complete tripartite 3-graph with classes $V_1,V_2,V_3$ by removing a linear⁴ 3-graph with the additional property that every edge contains one element from each of the classes $V_1$, $V_2$ and $V_3$. It is not hard to show that every 3-graph in $Q(n)$ is in fact induced $\{K_4^3,E_1\}$-free and that $|Q(n)|=2^{\mathrm{\Omega }(n^2\log n)}$ (see the proof of Theorem 1.2 in Section 4). Balogh and Mubayi [7] conjectured that almost all induced $\{K_4^3,E_1\}$-free 3-graphs are in this family, up to isomorphism.

Conjecture 1.1 Balogh and Mubayi [7]

Almost all induced $\{K_4^3,E_1\}$-free 3-graphs on $[n]$ are in $Q(n)$, up to isomorphism.

The motivation behind this conjecture comes from similar results. In particular, Person and Schacht [29] proved that almost all Fano-plane free 3-graphs are bipartite, and Balogh and Mubayi [6] proved that almost all $F_5$-free triple systems are tripartite, where $F_5$ is the 5-vertex 3-graph with edge set $\{123,124,345\}$. See also [5] for results along the same line.

The problem of counting r-graphs which are free of forbidden structures dates back to the work of Erdős, Kleitman and Rothschild [14] in the context of graphs. They showed that the number of $K_s$-free graphs on n vertices is $2^{(1+o(1))\mathrm{\text{ex}}(n,K_s)}$. Their work was later extended to all non-bipartite graphs by Erdős, Frankl and Rödl [13] using the Szemerédi regularity lemma. For other related results, see [8], [9], [13], [16], [21], [28], [36]. In a sequence of papers [31], [32], [33], Prömel and Steger studied the corresponding problem for induced graphs. Their results were stated in terms of a different notion of extremal number, which was latter generalized by Dotson and Nagle [12] as follows. Given a family of r-graphs $\mathcal{F}$, let M and N be r-sets⁵ in $[n]:=\{1,\dots ,n\}$ with the following properties: (i) $M\cap N=\varnothing$; and (ii) for $G\subseteq \left(\begin{array}{c}[n]\\ r\end{array}\right)\setminus (M\cup N)$, the r-graph $G\cup M$ is induced $\mathcal{F}$-free. The notation $\left(\begin{array}{c}[n]\\ r\end{array}\right)$ stands for $\{S\subseteq [n]:|S|=r\}$. The ⁎-extremal number ${\mathrm{\text{ex}}}^{⁎}(n,\mathcal{F})$ is defined as

$${\mathrm{\text{ex}}}^{⁎}(n,\mathcal{F}):=\left(\begin{array}{c}n\\ r\end{array}\right)-\underset{M,N}{\min }(|M|+|N|),$$

where the minimum is over all r-sets $M,N\subseteq [n]$ satisfying (i) and (ii). In 1992, Prömel and Steger [32] showed that the number of induced F-free graphs on n vertices is $2^{{\mathrm{\text{ex}}}^{⁎}(n,\mathcal{F})+o(n^2)}$. This result was later extended by Alekseev [1] and Bollobás and Thomason [10] for families of graphs, and by Kohayakawa, Nagle and Rödl [22] for 3-graphs. In 2009, Dotson and Nagle [12] generalized these results, showing that for all families of r-graphs $\mathcal{F}$ the number of induced $\mathcal{F}$-free r-graphs is $2^{{\mathrm{\text{ex}}}^{⁎}(n,\mathcal{F})+o(n^r)}$.

For a family $\mathcal{F}$ of r-graphs such that ${\mathrm{\text{ex}}}^{⁎}(n,\mathcal{F})=o(n^r)$, the counting results mentioned above are not precise. In the case of graphs, Alon, Balogh, Bollobás and Morris [2] obtained a more refined result. They showed that the number of induced $\mathcal{F}$-free graphs on n vertices is $2^{{\mathrm{\text{ex}}}^{⁎}(n,\mathcal{F})+O(n^{2-\epsilon })}$, where $\epsilon >0$ depends only on the family $\mathcal{F}$. Terry [37] generalized this result to finite relational languages which in particular covers r-graphs. For a family of r-graphs $\mathcal{F}$, her result says that the number of induced $\mathcal{F}$-free 3-graphs is either $2^{\mathrm{\Theta }(n^r)}$ or there exists $\epsilon >0$ such that for all large enough n, the number of induced $\mathcal{F}$-free 3-graphs is at most $2^{n^{r-\epsilon }}$.

Our first theorem determines the number of induced $\{K_4^3,E_1\}$-free graphs up to a constant factor on the exponent, making progress towards Conjecture 1.1.

Theorem 1.2

The number of induced $\{K_4^3,E_1\}$-free 3-graphs on n vertices is $2^{\mathit{\Theta }(n^2\log n)}$.

More generally, we also determine the number of induced $\mathcal{F}$-free 3-graphs on n vertices for all families $\mathcal{F}$ of 3-graphs on 4 vertices. Since every 3-graph on 4 vertices is determined by its number of edges, our result is stated in terms of forbidden number of edges. For a set $L\subseteq \{0,1,2,3,4\}$, let $f(n,3,4,L)$ be the number of 3-graphs on n vertices which do not induce $i\in L$ edges on any set of 4 vertices. Our result can be stated as follows, where we do not attempt to optimize the constants in the exponent.

Theorem 1.3

Let $L\subseteq \{0,1,2,3,4\}$ be a set. Then, the following holds for $n\ge 13$.

• If $\{0,4\}\subseteq L$ or $\{1,2,3\}\subseteq L$, then $f(n,3,4,L)\in \{0,1,2\}$;

• If $L=\{0,2,3\}$ or $L=\{1,2,4\}$, then $f(n,3,4,L)=n+1$;

• If $L=\{0,1,3\}$ or $L=\{1,3,4\}$, then $f(n,3,4,L)=2^{\mathit{\Theta }(n\log n)}$;

• If $L=\{1,3\}$, then $f(n,3,4,L)=2^{\left(\begin{array}{c}n-1\\ 2\end{array}\right)}$;

• If $L\in \{\varnothing ,\{0\},\{1\},\{3\},\{4\},\{0,1\},\{3,4\}\}$, then $f(n,3,4,L)=2^{\mathit{\Theta }(n^3)}$;

• For all the remaining cases, we have $f(n,3,4,L)=2^{\mathit{\Theta }(n^2\log n)}$.

Note that some of the statements in Theorem 1.3 are trivial and others are known. We included those for the sake of completeness. For a list $L\subseteq \{0,1,2,3,4\}$, define $L^c=\{4-i:i\in L\}$. It is not hard to check that if a list L belongs to item (f), then some $T\in \{L,L^c\}$ satisfies $T=\{1,4\}$ or $\{2\}\subseteq T\subseteq \{0,1,2\}$. See Table 1 for a detailed list with the bounds on $f(n,3,4,L)$ and the references of statements which proves each of them.

Before we state our next theorem, we need some notation. Let $k,r\in \mathbb{N}$ and $L\subseteq \{0,1,\dots ,\left(\begin{array}{c}k\\ r\end{array}\right)\}$ be a set, which we refer to as a list. We say that an r-graph G is $(L,k)$-free if for all $i\in L$ there is no set of k vertices in G inducing exactly i edges. By generalizing our previous notation, we denote by $f(n,r,k,L)$ the number of $(L,k)$-free r-graphs on n vertices. Our next theorem extends Theorem 1.2 to r-graphs and 3-good lists. We say that a list L is 3-good if $\{i,i+1,i+2\}\cap L\ne \varnothing$ for all $i\in \{0,1,\dots ,\left(\begin{array}{c}k\\ r\end{array}\right)-2\}$. That is, the complement of L does not contain three consecutive integers. Throughout this paper, all logarithms are in base 2.

Theorem 1.4

Let $n\ge k>r\ge 2$ be integers and $L\subseteq \{0,1,\dots ,\left(\begin{array}{c}k\\ r\end{array}\right)\}$ be a list. If L is 3-good, then

$$f(n,r,k,L)\le 2^{2k{n}^{r-1}+n^{r-1}\log n}.$$

The main tool behind the proof of Theorem 1.4 is a lemma which counts the solutions of a certain constraint satisfaction problem, see Lemma 3.1. For $L=\{2,3,\dots ,r+1\}$, we observe that $f(n,r,r+1,L)$ is equal to the number of r-graphs such that, for every pair of edges, the size of their intersection is not $r-1$. This is related to the problem of counting designs, a heavily studied object in combinatorics, see [20], [24], [25].

The rest of this paper is organized as follows. In Section 2 we discuss the sharpness of Theorem 1.4; in Section 3 we present the proof of Theorem 1.4; in Section 4 we prove Theorem 1.2, Theorem 1.3.