One of the classical problems in extremal graph theory is that of finding the maximum density of a graph which does not contain some fixed graph . This density is known as the Turán density for and is defined as , where the Turán number is the maximum number of edges in a graph on vertices which does not have as a subgraph. Mantel proved that and later Turán gave a complete answer for [Tur41].
Motivated by a question of Erdős regarding the maximum density of a triangle free subgraph of a graph [BSTT06] investigated a modified version of the Turán density, namely that of finding the maximum density of a 3-partite graph which does not contain a . The problem was given a complete solution in terms of the three bipartite graphs induced by each pair of vertex classes of the 3-partition. Later [BJT10] investigated the number of s as a function of these densities, and a sharp result was given for large enough densities. Enumeration of triangles in general graphs has a long history and was finally solved by Razborov [Raz08].
Our aim in this paper is to investigate generalisations of these questions to uniform hypergraphs. In particular we will determine the maximal density of an -partite -uniform hypergraph which does not contain the complete hypergraph . We will also present a sharp bound on the number of copies of . These results demonstrate a qualitative difference between graphs and hypergraphs with , where interestingly enough the extension to the hypergraph case is less complex than the graph case. The existence condition found in [BSTT06] for graphs is non-linear in terms of the involved densities, as are the counting results from [BJT10], but as our results show the corresponding condition, and counting function, for are given by simple linear functions.
For hypergraphs far less is known in the non-partite case than for graphs. Turán conjectured that , and gave a matching construction for the lower bound. Using flag-algebra Razborov [Raz10] has proven that . For Giraud [Gir90] gave a construction which shows that , and Sidorenko [Sid95] conjectured that this is in fact the correct value. The best current upper bound was given by the first author in [Mar09]. For much less is known. De Caen [dC83] proved that
, and this was later sharpened somewhat for oddin [CL99] and even in [LZ09]. As a corollary to one of our results we will get a short proof of de Caen’s bound.
Mubayi and Talbot [MT08] investigated the global density of a 4-partite 3-graph such that . and proved the sharp result .
One of the few counting results for hypergraphs is by Mubayi [Mub13] who used the hypergraph removal lemma [Gow07, NRS06, RS06, Tao06] to prove lower bounds on the number of copies of in a hypergraph with edges. These bounds apply to certain forbidden hypergraphs with the property that there is a unique -free hypergraph on edges, and the bounds are of the form , where is the minimum number of copies of in any -vertex -graph on edges.
For we will refer to an -uniform hypergraph as an -graph. An -graph is -partite if its vertex has a partition in vertex classes , such that each edge has at most one vertex in any class .
Given a -partite -graph with vertex classes and a set of indices we let denote the -partite -graph induced by the classes with indices not in . If then we abbreviate this as
We define the weight of an edge to be the product , and the weight of a set of edges as the sum of their weights.
An unweighted -graph will here be seen as the same graph with a weight function for all vertices .
Given an -partite -graph we let be the density of the -partite graph induced by , which is , and
is the vector of these densities.
2 Threshold and the minimum number of in an -partite -graph
Given an -partite -graph we want to count the number of s it contains. It will be convenient to do this in terms of the density of s
Given an -partite -graph we define the density of s in to be
We define the minimum density of s as
where the infimum is taken over all -partite -graphs with density .
We will need the following polynomial from [BJT10].
The poof of the following lemma is a routine, but somewhat lengthy, calculus exercise.
If are real numbers in the interval and then .
In particular if each if each of and is at least then .
If is a vector with rational numbers between 0 and 1, and there exists such that
then there exists a weighted -partite -graph with rational weights such that is the density vector of and .
We will prove the statement by induction on . For the statement is true by Theorem 2.4 of [BJT10]. By that theorem there is an optimal graph which is a weighted 3-partite graph on six vertices whose 3-partite complement is a perfect matching. Hence we only need to prove the statement for .
Without loss of generality we assume that for and hence, by averaging, that . Thus, by Lemma 2.3, we know that .
We now assume that the statement is true for . In order to build an -graph which satisfies the statement in the Lemma we take an -partite -graph , with the first parts of as its density vector, which satisfies the statement for , and build an -partite -graph by adding one new class which contains a single vertex , with weight 1.
For every edge in we let be an edge of . This means that for the density contributed to by these edges will be the same as in . Hence has the desired density for these classes, and the edges added so far do not give rise to a .
We will now add edges among the first classes. One of these edges will be part of a weighted in if the corresponding -tuple in is a in , and by induction we know the density of such .
As pointed out in [BJT10] we may assume that is in fact a simple unweighted hypergraph, since a hypergraph with rational weights can be modified by a suitable blow-up into an equivalent unweighted hypergraph on a larger number of vertices.
In the density of partite -tuples which do not span a is , and all these tuples can be added as edges without creating a in . Each additional edge added after these form a unique together with the vertex in , so if we add enough edges to reach the desired density we will have
density of in . ∎
Our next step will to prove a lower bound for the density of s.
Let be an -partite -graph, then the density of s satisfies the following inequalities
If satisfies the conditions of Lemma 2.4 then
Let , where is a set of size , be 1 if is an edge of and 0 otherwise, let be 1 if is the vertex set of a contained in , and 0 otherwise.
But the left hand side is the sum of the densities in and the right hand side is plus the density of s so
and so , and we have proven part 1.
If all densities in then
If , for all and then there exists with these densities such that
We note that the graph which achieves the minimum number of copies of s is not unique, i.e.the -partite Turán-problem for is not stable. This is true already for , and Lemma 2.4 gives distinct extensions to higher values of .
3 Balanced Codegrees
Our second result concerns degrees rather than densities, and for -graph we have found it natural to consider the degrees of the -tuples of vertices in .
Given a multipartite -graph , with partitioned into at least classes, we say that a -tuple of vertices from is partite if it has at most one vertex in each class of .
Given a partite -tuple we say that an -tuple is completing if . We call the set of completing -tuples for the neighbourhood of . The neighbourhood of in a set of classes is the set of completing -tuples in , and is denoted
The degree of is and the degree in an -tuple of classes is .
By the minimum codegree of an (multipartite) -graph we refer to the minimum degree of all (partite) -tuples of vertices in
We say that a partite -tuple of vertices in has strictly balanced degree if it has the same number of completing -tuples in each of the -tuples of classes which does not intersect.
If the partite -tuples of have strictly balanced degrees and then .
Note that the condition on the partite -tuples means that such a tuple splits its neighbourhood equally between each of the two parts which the tuple does not intersect, but the sizes of those neighbourhoods may differ between different tuples.
We will first look at the case . Let be an edge of and let be the -tuples which are subsets of . If does not contain we must have that
since otherwise there would be a common vertex in the neighbourhoods of the -tuples, and we would have a .
If we sum over all edges in we find that
where the inner sum is over all partite -tuples .
Using the fact that each has strictly balanced degree we rewrite this as
Using the Cauchy-Schwarz inequality we get
Substituting this we find that
But is independent of so we can divide both sides by to get
The results follows for other values of in the same way.
As shown in [BSTT06] this result is sharp for , but we do not have a matching lower bound for larger .
Relating to the case where all densities are equal we get the following
If all partite -tuples of have strictly balanced degree and for all then .
This can be compared to the minimum codegree which forces a in the non-partite case. In [LM14] it was proven that there are -graph with minimum -degree which do not contain a , and it was conjectured that this is an optimal bound. This in turn implies that the global density is at least .
Another simple corollary of this result is de Caen’s upper bound on the Turán-density of .
Corollary 3.6 (de Caen [dC83]).
If is an -graph and then .
Given a labelled -graph with vertex set we will form a new -partite -graph . The vertex set of is the disjoint union of copies of .
Given an edge of , a choice of of the classes , and a permutation , we let , where the th vertex in the tuple is the vertex in the th of the vertex classes, be an edge in the -partite subgraph of induced by those classes. Note that in this way each edge of gives rise to edges in each of the -partite subgraphs of . (So if we look at ordinary graphs and let be a single edge (1,2) then would become a 6-cycle which winds twice through the three parts.)
The -graph has strictly balanced codegrees, since the number of neighbours in , of a partite -tuple in , only depends on whether intersects or not, and the number of neighbours of in .
By the assumptions each in is strictly greater than , so contains a copy of , and the corresponding vertices form a in . ∎
In fact the proof of Theorem 3.4 can easily be modified to give a bound for the degree required to give a copy of , the -graph obtained by deleting any edges from , we just need to modify the first bound for to be less than . This gives
If the partite -tuples of have strictly balanced degrees, , and then .
For the graph is in fact -partite, and the strictly balanced degree condition means that if has at least one edge then any edge will be part of a , which is simply two edges overlapping on an -tuple.
If all partite -tuples of have balanced degree, , and for all then .
If is an -graph, , and then .
The case of the latter result was proven by de Caen in [dC83].
4 Open problems
Following [BSTT06] and the results in this paper concerning the codegree it is natural to ask what happens for general degrees in -graphs with . The first open case is vertex degrees for -graphs
Let be a 4-partite -graph with balanced vertex degrees. Which densities force a in ?
For the corresponding question is open for balanced -degrees for all .
Let be an -partite -graph with . Which densities forces a if the partite -tuples have balanced degrees, where .
Let be a -partite -graph with all densities at least . Given an -graph , we can ask for the minimum density which forces a copy of in . We call this density .
Does there exist a finite such that for ?
For and the answer is yes by [Pfe12].
This research was done while the authors were attending the research semester Graphs, hypergraphs and computing at Institut Mittag-Leffler (Djursholm, Sweden). The first author was supported by The Swedish Research Council grant 2014–4897. The second author was supported by ERC Advanced Grant GRACOL.
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