The number of hypergraphs without linear cycles
The -uniform linear -cycle is the -uniform hypergraph on vertices whose edges are sets of consecutive vertices in a cyclic ordering of the vertex set chosen in such a way that every pair of consecutive edges share exactly one vertex. Here, we prove a balanced supersaturation result for linear cycles which we then use in conjunction with the method of hypergraph containers to show that for any fixed pair of integers , the number of -free -uniform hypergraphs on vertices is , thereby settling a conjecture due to Mubayi and Wang.
2010 Mathematics Subject Classification:Primary 05D10; Secondary 05D40
The general problem of asymptotically enumerating discrete structures with various forbidden substructures has a very rich history. The simplest such question that one may ask is as follows: given a fixed graph , how many -vertex graphs are there that contain no copy of ? In the case where the fixed forbidden graph is not bipartite, reasonably precise estimates are available from the work of Erdős, Frankl and Rödl []; see also []. On the other hand, the case where the fixed forbidden graph is bipartite remains an active area of investigation since the corresponding ‘Turán problem’ of determining the maximum number of edges in an -vertex -free graph remains open in general; consequently, much of the work in this case has been focused on understanding the behaviour of important prototypical examples such as complete bipartite graphs (see [, ]) and even cycles (see [, ]). Of course, one may ask similar questions for various other discrete structures; see [] for a broad overview of the area.
In this paper, we shall be concerned with enumerating uniform hypergraphs. For an integer , an -uniform hypergraph (or -graph for short) is a pair of finite sets, where the edge set is a family of -element subsets of the vertex set . For a fixed -graph and a natural number , the corresponding Turán problem asks for the determination of , the maximum number of edges in an -vertex -graph that contains no copy of as a subgraph, and the associated enumeration problem asks for the determination of , where denotes the family of all -free -graphs on the vertex set . The Turán problem and the enumeration problem for a given -graph are closely related; indeed, for a fixed -graph , we trivially have
We remark that it is generally believed that the lower bound in (1) is closer to the truth (provided is not ‘degenerate’), and indeed, all existing results in the area support this belief.
Mirroring the situation described earlier for graphs, for each , the enumeration problem for a fixed forbidden -graph has a reasonably satisfactory solution in the case where is not -partite. Indeed, it follows from the work of Nagle, Rödl and Schacht [] on hypergraph regularity that for any fixed -graph , we have
since for any -graph that is not -partite, the above bound complements the trivial lower bound in (1).
However, the enumeration problem for a fixed forbidden -partite -graph is poorly understood; indeed, for any such , we know that for some constant depending on alone, so the upper bound from (2) is some ways off from the trivial lower bound in (1). Consequently, as in the case of graphs, it is important to understand the behaviour of prototypical examples of -partite -graphs. Here, following Mubayi and Wang [], we shall investigate the enumeration problem for one such prototypical family of -partite -graphs, namely, the family of -uniform linear (or loose) cycles.
For integers and , the -uniform linear -cycle is the -graph on vertices whose vertices can be ordered cyclically in such a way that the edges are sets of consecutive vertices in this ordering such that every two consecutive edges share exactly one vertex. It is known from the work of Füredi and Jiang [] and of Kostochka, Mubayi and Verstraëte [] that for any fixed , we have ; it then follows from (1) that we trivially have
for any fixed , and also established this conjecture in the case where and is even; they also showed that the trivial upper bound in (3) is not sharp for general and , and their improvements over the trivial upper bound were subsequently refined by Han and Kohayakawa []. Here, we shall establish (4) in full generality, thereby resolving the conjecture of Mubayi and Wang []; our main result is as follows.
For every pair of integers , there exists such that
for all .
Our proof of Theorem 1.1 proceeds by iteratively applying a construction of ‘hypergraph containers’ introduced independently by Balogh, Morris and Samotij [] and by Saxton and Thomason []. To make use of the framework of hypergraph containers, we prove a ‘balanced supersaturation’ theorem for linear cycles: this result roughly states that an -graph on vertices with significantly more than edges contains many copies of which are additionally distributed relatively uniformly over the edges of ; as remarked in [], this result might be of independent interest.
A brief word on how our approach to proving Theorem 1.1 compares to the approach adopted by Saxton and Morris [] to count graphs without even cycles is perhaps in order. At a very high level, both proofs are based on combining an appropriate balanced supersaturation result with the method of hypergraph containers; however, the arguments used to establish balanced supersaturation are qualitatively very different in the two cases. Indeed, as is evident from [], the problem of enumerating graphs without copies of even cycles more resembles the problem of enumerating -free linear hypergraphs than the problem at hand here.
This paper is organised as follows. We set up some notation and collect together the results we need for the proof of our main result in Section 2. We state and prove our ‘balanced supersaturation’ theorem for linear cycles in Section 3, and then demonstrate how to deduce Theorem 1.1 from this result in Section 4. We conclude with some discussion of open problems in Section 5.
For , we denote the set by . For a set , we write for the set of all subsets of , and given , we write for the family of -element subsets of . In this language, an -graph is a pair of finite sets with ; also, as is customary, we shall always refer to -graphs as graphs.
Let be an -graph. For a set with , we define to be the set of edges of containing , i.e.,
and we define its degree by . Next, for , we define the maximum -degree of by
Also, we define the average degree of by . Finally, for , the co-degree function of is given by
Given an -graph , we write for the subgraph of induced by a set . A subset of the vertex set of an -graph is said to be independent in if no edge of is contained in ; equivalently, is independent in if is empty. We shall make use of the following hypergraph container theorem; see [], for example.
For each , there exist positive constants , and such that the following holds for all . For each and each -vertex -graph , if is such that , then there exists a family of at most
subsets of (called the containers of ) such that
for each independent set , there exists a container such that , and
for each container . ∎
Next, we introduce some notation for working with linear cycles. For integers and , recall that the -uniform linear -cycle is the -graph on vertices whose vertices can be ordered cyclically in such a way that the edges (of which there are precisely ) are sets of consecutive vertices in this ordering such that every two consecutive edges share exactly one vertex. The core of an -uniform linear -cycle is the set of those vertices of the cycle which are each contained in two edges of the cycle. For , a skeleton of an -uniform linear -cycle is a triple , where is an ordering of the edges of the cycle, is an ordering of the core of the cycle, and is a choice of non-core vertices of the cycle such that
for each with the convention that , and
for each .
Given a skeleton as above, we call the set the support of the skeleton, and we call the graph on the support depicted in Figure 1 the canonical triangulation of the (support of the) skeleton.
We shall use skeletons and their canonical triangulations to help simplify the bookkeeping when counting the number of copies of in an -graph. We begin with the following simple observation.
An -uniform linear -cycle possesses exactly different skeletons. ∎
Next, we observe the following property of canonical triangulations.
For any proper subset of the support of a skeleton inducing at least one triangle in the corresponding canonical triangulation, there exists a vertex in the support with the property that there exist such that the sets and both induce triangles in the canonical triangulation.
The claim follows from the simple observation that for each triangle of the canonical triangulation, there exists an ordering of the remaining vertices of the support such that for each , there exist such that the sets and both induce triangles in the canonical triangulation. ∎
Finally, to prove Theorem 1.1, we shall need the existence of graphs satisfying a very mild regularity condition; in particular, we shall make use of the following simple fact.
For each and each , there exists an -vertex graph with the property that each vertex of the graph has degree either or . ∎
A few more remarks about notation are in order before we proceed. We shall make use of standard asymptotic notation; in the sequel, constants suppressed by the asymptotic notation are allowed to depend on the fixed parameters and . For the sake of clarity of presentation, we systematically omit floor and ceiling signs whenever they are not crucial.
3. Balanced supersaturation
The purpose of this section is to prove the following balanced supersaturation result for linear cycles which asserts that in any -graph on with significantly more than edges, one can find many copies of that are additionally ‘well-distributed’ across the -graph in question.
For every pair of integers , there exists such that the following holds for all . Given an -graph on with for some , there exists a -graph on , where each edge of is a copy of in , such that
for each ,
where if and if .
We start by setting and ; note that since if and if , we have if and if .
We shall construct three hypergraphs on from : an -graph which is a subgraph of , a -graph and a graph . First, we obtain from by repeatedly applying, until it is no longer possible to do so, the following deletion rule: if there exists either a -set with or a -set with , delete every edge of containing the corresponding set. We then define to be the -graph whose edge set consists of those triples such that , and analogously define to be the graph whose edge set consists of those pairs such that .
We observe that , and have the following properties by virtue of how they are constructed.
First, for each and for all , and analogously, for each and for all
Next, we have for each . Indeed, if this fails to hold for some , then
Finally, we have . Indeed, the number of edges deleted from to obtain is at most
in the case where , and at most
in the case where .
We shall construct from the copies of in using and ; to do so it will be helpful to define some auxiliary structures. First, we define a collection of graphs on , one for each , as follows. From (B), we know that for each , so we may appeal to Proposition 2.5 and fix a graph on with the property that each has degree either or in . Next, let be the graph on whose edge set is the union of the edge sets of the graphs over ; since has positive degree in if and only if , it follows that the degree of each in is at most . Finally, we know from (A) that for each ; we may therefore fix a subset of size for each .
We shall construct by specifying a skeleton (though possibly more than one) for each copy of in that we wish to include in . Furthermore, we shall guarantee that the support of each skeleton that we specify has the following adjacency property: if and are two subsets of the support that induce triangles in the corresponding canonical triangulation, then and are both edges of that are adjacent in the graph .
We now describe an algorithm to construct skeletons of copies of in whose supports additionally satisfy the adjacency property; recall that specifying a skeleton of a copy of in involves specifying a triple , where is an ordering of the edges of the cycle, is an ordering of the core of the cycle, and is a choice of non-core vertices of the cycle.
We start by choosing an edge of , in ways, and setting . We then choose three distinct vertices from and designate these vertices to be , and in some order.
Next, we specify inductively as follows. Having specified and , we first fix as follows. Consider the triple , where if and if , and the pair . Let be a neighbour of in the graph chosen in such a way that is distinct from all the already specified vertices of the support and so that . We then set , noting that since has at least neighbours in , there are at least choices for . If is odd and it so happens that , then we stop after fixing . If not, then we pick in a manner analogous to how we chose , working instead with the triple and the pair . If is even and it so happens that , then we stop after fixing . Observe that these choices ensure that the subset of the support specified so far satisfies the adjacency property.
Now, we finish specifying the support of the skeleton by inductively choosing as follows. For , having specified, , we fix as follows. Note that there exists a unique vertex such that the triple induces a triangle in the canonical triangulation; let and . Let be a neighbour of in the graph chosen in such a way that is distinct from all the already specified vertices of the support and so that . We then set , noting that since has at least neighbours in , there are again at least choices for . Again, note that these choices ensure that the support satisfies the adjacency property.
Finally, we finish specifying the skeleton by fixing inductively as follows. For , having specified, , we fix as follows. Let and consider the set of edges in containing . Choose an edge with the property that is disjoint both from and from the support of the skeleton. We then set , noting that the number of choices for is at least
in the case where , and trivially at least in the case where (since specifying the support specifies the entire skeleton when ).
We define by setting and including a copy of in in if and only if at least one skeleton of the cycle in question is generated by the above algorithm.
We now show that satisfies the requisite degree conditions. We remind the reader that constants suppressed by the asymptotic notation in what follows are allowed to depend on the fixed parameters and . We start by bounding the average degree of from below.
By Proposition 2.3, a fixed copy of in possesses distinct skeletons; therefore, it suffices to show that the number of distinct skeletons generated by the algorithm described above is . To do this, we count the number of distinct choices available to us at each stage in the above algorithm. Indeed, in the first stage (i), we have at least distinct choices, in the second stage (ii), we have at least distinct choices, in the third stage (iii), we have at least distinct choices, and in the final stage (iv), we have at least distinct choices. We know from (C) that , so we have
the claim follows since
To finish the proof, we bound the maximum degrees of from above.
For each , we have .
Fix and a set of size . Our aim is to show that ; to do this, we shall bound from above the number of distinct skeletons generated by our algorithm that contain , i.e., with the property that . Observe that we may assume that , for if not, the number of such skeletons, and consequently , is zero.
First, we fix which edges of the skeleton correspond to which edges in ; this may be done in at most ways. Next, for each edge in , we identify the two core vertices and one non-core vertex from that edge that belong to the support of the skeleton and note that this may be done in at most ways; let denote the subset of the support contained in some edge in and note that since two edges of an -uniform linear -cycle intersect in at most one core vertex, and since each core vertex belongs to precisely two edges, we must have .
We first estimate the number of ways of choosing, from , the vertices in the support of the skeleton not in . By repeatedly applying Proposition 2.4, we know that it is possible to find an ordering of the vertices of the support not in with the property that each for each , there exist such that the sets and both induce triangles in the canonical triangulation. We now count the number of ways to choose, for , the vertex from . Having picked , we know from the previous observation, and from the adjacency property of the support guaranteed by our algorithm, that there exist two triples that are adjacent in such that and . Consequently, the number of choices for is at most the number of ways of choosing multiplied by the maximum degree of , which is at most . It follows that the number of ways of choosing the support of a skeleton containing is .
Next, having fixed the support of a skeleton containing , we need to choose the remaining edges of the skeleton. The number of ways of doing this is easily seen to be at most .
Finally, combining the above estimates, we see that
this establishes the claim. ∎
4. Proof of the main result
For every pair of integers , there exists such that the following holds for all . Given an -graph on with for some , there exists a collection of at most
subgraphs of such that
each -free subgraph of is a subgraph of some , and
for each .
Now, fix and , where we define , with the benefit of hindsight, by
Since , we may apply Theorem 3.6 to to get a -graph on , where each edge of is a copy of in , such that
for each .
First, note that if is a -free subgraph of , then is an independent set in . Next, from the above bounds and the fact that , it is easily verified that the co-degree function of satisfies
We may therefore apply Theorem 2.2 to the -graph to get a collection of at most
subgraphs of such that
each -free subgraph of is a subgraph of some , and
for each .
To finish the proof, we proceed as follows. First, since and for all , it plainly follows that
Therefore, we have . Next, we know that each satisfies ; we claim that this implies that . To see this, note that
so if , then since and , we have
which is a contradiction. It follows that is indeed the required collection of subgraphs of . ∎
We are now in a position to prove Theorem 1.1.
Proof of Theorem 1.1.
Let be as promised by Proposition 4.9. We wish to estimate the number of -free -graphs on ; equivalently, we wish to estimate the number of -free subgraphs of the complete -graph on . To this end, we define a sequence of positive reals, and a sequence of families of -graphs as follows. We set and define , with being the first term of this sequence to satisfy . We take to consist of a single -graph on , namely, the complete -graph on , and for , we obtain from by replacing each -graph for which by the collection of its subgraphs guaranteed by Proposition 4.9.
Now, let . It is clear that each -free -graph on is a subgraph of some . Furthermore, it is easy to see that for each . Finally, a simple induction using (5) shows that
where the last inequality holds on account of the sequence decreasing geometrically.
It follows that the number of -free -graphs on is bounded above by
the result follows, with room to spare, by setting . ∎
Our results in this paper raise the natural question of deciding, for a fixed pair of integers , if it is the case that
as . While it is plausible that such an estimate is true, we conclude by warning the reader that an analogous estimate for the -uniform -cycle was shown not to hold by Saxton and Morris [].
The first and third authors were partially supported by NSF Grant DMS-1500121 and the first author also wishes to acknowledge support from an Arnold O. Beckman Research Award (UIUC Campus Research Board 15006) and the Langan Scholar Fund (UIUC).
Some of the research in this paper was carried out while the first author was a Visiting Fellow Commoner at Trinity College, Cambridge and the third author was visiting the University of Cambridge; we are grateful for the hospitality of both the College and the University.
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