Erdős-Ko-Rado for random hypergraphs: asymptotics and stability
We investigate the asymptotic version of the Erdős-Ko-Rado theorem for the random -uniform hypergraph . For , let and . We show that with probability tending to 1 as , the largest intersecting subhypergraph of has size , for any . This lower bound on is asymptotically best possible for . For this range of and , we are able to show stability as well.
A different behavior occurs when . In this case, the lower bound on is almost optimal. Further, for the small interval , the largest intersecting subhypergraph of has size , provided that .
Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in , for essentially all values of and .
The Erdős-Ko-Rado theorem  is a cornerstone in extremal combinatorics. Let denote the set , and denote the set of all -element subsets of . A family of -element sets is called a -uniform hypergraph on the vertex set , and such a hypergraph is called intersecting if holds for every edge . The Erdős-Ko-Rado theorem then states that for , an intersecting family must satisfy . This is best possible, as seen by the principal hypergraphs , which consist of all edges containing the fixed element .
We investigate a random analogue of the Erdős-Ko-Rado theorem in which the ambient space in the theorem is replaced by a random space. Random analogues of extremal results have been studied extensively in the last decades, and we refer to [24, 8, 6, 23] for the history of this line of research and recent breakthroughs.
The ambient random space we will work with is , the binomial random -uniform hypergraph on the vertex set in which each edge is included in independently with probability . Further, for a -uniform , let denote the size of the largest intersecting subhypergraph of , i.e., . In this notation the Erdős-Ko-Rado theorem states that .
All asymptotic limits in this paper are taken as . If we write or , it means that . In particular, the notation represents a function that goes to as , as usual. For simplicity, we omit floor and ceiling functions, whenever they are not essential. We say that a sequence of events holds asymptotically almost surely if as . By we denote .
We will be interested in for and all . This question was investigated by Balogh, Bohman and Mubayi , which obtained very precise results on the size and on the structure of the largest intersecting family in , for . For larger , they obtained asymptotic tight bounds on , however, only for rather large values of . In general, their result highly depends on the range of and , and hence it is slightly cumbersome to state. Therefore, we will only partially discuss it here, and refer to  for detailed information. Their result concerning the large range of is given below in Proposition 1.1.
Proposition 1.1 (Proposition 1.3 in ).
Let and . If and , then almost surely as :
In other words, for this range of , the expected size of the intersection of a principal family with is very close to the size of a maximum intersecting subfamily of . We extend this result, and provide an almost complete description of as follows.
For all there exists a constant for which the following holds. Let , , where , , and . Then almost surely as :
if and ,
The first bound follows from a standard deletion argument, and we state it here for completeness. Note also that is monotone in , hence, in the range of around not mentioned in the theorem, we have due to (4).
If is linear in , the bounds in (1) and (4) determine asymptotically for essentially all . Here, we have a change of behaviour around . Roughly speaking, for below , essentially all of is intersecting. Beyond that point, i.e. for , the largest intersecting subhypergraph of has size very close to the size of the intersection of a principal hypergraph with . Observe that cases (2) and (3) are trivial for .
For there is a rather short range of where reveals a “flat” behaviour. Indeed, the upper bound (3) shows that grows slowly with , since it appears only in the -term. The corresponding lower bound in (2) shows that for this bound is tight up to a multiplicative constant. We provide no lower bound for the range here, as in this case the result of Balogh et al. is more satisfactory. Again, we refer to  for further information.
Although the “flat range” phenomenon might come as a surprise, it has been observed elsewhere. Indeed, in the dense case, i.e. for , and for , the size of the largest intersecting family is vanishing compared to the ambient space, that is, . For these so called “degenerate” problems, the random analogues typically reveal such an intermediate flat behaviour, as observed for example in [19, 20].
The question of for which range of the largest intersecting family is indeed the projection of a principal family has been successfully addressed in  for . For larger , which we are mainly interested in, the problem seems to be more complicated, and has only been studied recently in , for constant . We make no contribution to this question here. However, besides the bounds on , we are able to show stability for in the same range for as in case (4) in Theorem 1.2.
For every and there exist constants and for which the following holds. For any and , asymptotically almost surely stability holds, i.e., for every intersecting family of size , there is an element that is contained in all but at most elements of .
In the dense case, i.e. for , the result was proven by Friedgut . Indeed, the proof of Theorem 1.3 relies on the result of Friedgut and on a removal lemma for the Kneser graph due to Friedgut and Regev .
Further results and organization.
In proving Theorems 1.2 and 1.3, it will be convenient for us to work with the Kneser graph . The vertex set of this graph is , and two -element sets form an edge if and only if they are disjoint. Hence is a -regular graph on vertices, and a hypergraph is intersecting if and only if is an independent set in . Further, let denote the subgraph of induced on the random vertex set obtained by including each vertex from independently with probability . Due to the correspondence, all bounds on intersecting subgraphs of will follow from corresponding bounds on the size of largest independent sets in .
Using this translation, Theorem 1.2 follows from a more general scheme which relies on the technical Proposition 2.6 and Lemma 2.2, to be introduced in the next section. Further, for Theorem 1.3 we will need Lemma 2.5, which together with Lemma 2.2 will be proven in Section 3. Based on these results, we will give the proofs of Theorems 1.2 and 1.3 in Section 2.
In general, the proof scheme based on Proposition 2.6 and Lemma 2.2 can be used to bound the size of the largest independent sets in random subgraphs of any -regular graph (actually, a sequence of graphs). Here by random subgraph we mean the graph induced on a binomial random subset of the vertex set. This application yields asymptotically sharp bounds if has an independent set of size (close to) . Indeed, Theorem 1.2 shows such an application to the Kneser graph, and there are many other graphs for which this applies. We refer, e.g., to  for a list of such graphs which include the weak product of the complete graph, line graphs of regular graphs which contain a perfect matching, Paley graphs, some strongly regular graphs, and appropriate classes of random regular graphs (see Section 5.1. of ).
The proof of Proposition 2.6 will be given in Section 4. It is based on a description of all independent sets in locally dense graphs. This idea can be traced back to the work of Kleitman and Winston , and has been exploited in various contexts since their work. Though similar proofs have been given elsewhere, none of them seems to fully fit in our context. This also applies to the powerful extension of the ideas of Kleitman and Winston to hypergraphs due to Balogh, Morris and Samotij in  (see also ), which only partially suits our needs.
As mentioned before, the proofs of the main theorems rely on Proposition 2.6. A central notion employed in this proposition which applies to is the following.
Given , , and a graph on vertices, we say that is ,-supersaturated if for any subset with , we have
In addition, let and . A sequence is called -supersaturated if is -supersaturated for each .
Hence, in a -supersaturated graph each set of size at least spans many edges. Indeed, up to the multiplicative factor , spans as many edges as expected from a random subset of of the same size.
Let be a -regular graph on vertices, and let denote the smallest eigenvalue of the adjacency matrix of . Then every set satisfies
As the eigenvalues of the Kneser graph are known due to Lovász , we immediately conclude the following supersaturation for the Kneser graph.
Let and . Then is -supersaturated.
Let . Let be a graph on vertices, and let be a family of sets. We say that is -stable with respect to if for every with , we have either
, for some .
In addition, let , be a sequence of graphs, and with be a sequence of families of sets. We say that is -stable if for any there exists and such that for all , the graph is -stable with respect to .
It is instructive to think of as the family of largest independent sets in , and of as the size of each . The first part of the definition roughly says that if is robustly stable, then any vertex set whose size is close to the size of a largest independent set in must either contain many edges, or be close to a largest independent set in structure.
The Kneser graph satisfies robust stability for linear in , as stated in the next lemma. It is a direct consequence of the corresponding stability result proven by Friedgut , and the removal lemma proven by Friedgut and Regev . Again, we refer to Section 3 for the details of the proof. In the following, let denote the principal hypergaph centered at , i.e., the hypergraph consisting of all -element subsets of containing .
Let and , where , and let . Further, let , and set . Then is -stable.
With supersaturation and robust stability defined, we are now ready to state our main technical result. Given a graph , we use to denote the size of the largest independent set in . Also, for a finite set , we let be a random subset of obtained by selecting each element independently with probability .
Let and be -valued functions, and let be a family of graphs, where each has vertices (with ) and average degree . For any constant there exist constants and such that for any probability sequence , the following holds. For a random spanning subgraph , where , we have:
If , then asymptotically almost surely.
If is -supersaturated and , then
If is -supersaturated and , then
If is -stable and , then with probability at least , the following holds: every independent set in of size at least satisfies for some .
Let be a sequence of graphs on vertices with average degree at most . If each is triangle-free, then contains an independent set of size . ∎
Finally, we shall repeatedly use Chernoff’s bound for binomial random variables, which we state here for reference (see [17, Theorem 2.1]).
Given integers and , we have:
Proof of Theorem 1.2.
Given , apply Proposition 2.6 with in order to obtain a corresponding constant . Let . Further, let , and . Recall that is a -regular graph on vertices, with and . Let , where , and is the set obtained by including each vertex of independently with probability . We apply Proposition 2.6 to , with functions and as defined above. The first bound of Theorem 1.2 follows immediately from the first case of Proposition 2.6.
For the third and fourth bounds of Theorem 1.2, note that by Lemma 2.3 applied with , we know that is -supersaturated, with and . Thus we can apply Proposition 2.6 in both cases. We start with the third bound of Theorem 1.2. Assume that , since for linear in this range of is trivial. By the second part of Proposition 2.6 applied with , we derive that for , which contains our interval for in the third case, we have
with probability at least . As , this probability tends to one as goes to infinity, which gives the upper bound in the third case.
Next we show the fourth bound of Theorem 1.2. The lower bound follows by considering the subhypergraph of consisting of all hyperedges containing, say, the element . Using the Chernoff bound (Lemma 2.8), we have with high probability that this (intersecting) subhypergaph has size at least . For the upper bound, we apply the third bound of Proposition 2.6 with and , as chosen above. Then, by the choice of , we have for , and the claim follows.
Finally, we prove the second bound of Theorem 1.2. Observe that this range of is nontrivial only if . By Chernoff’s bound, almost surely has at least vertices. Further, , and it is not hard to see that . By Chebyshev’s inequality, we derive
which goes to zero by the choice of .
For and , asymptotically almost surely the number of triangles in is at most .
The expected number of triangles in is at most . Using Markov’s inequality and , the claim follows if we can show that Indeed,
and using for , we obtain together with our assumption that
which completes the proof of the claim. ∎
Hence, by removing at most vertices, we obtain a triangle free graph with at least vertices, and no more than edges. Consequently, this graph has average degree at most , and due to Proposition 2.7, it contains an independent set of size
This completes the proof. ∎
Proof of Theorem 1.3.
Let be fixed, and . Again, let denote the Kneser graph . Set , and for a given , let be the set of all principal hypergraphs , for . By Lemma 2.5, the family is -stable, where . For a given , we apply Proposition 2.6 in order to obtain constants and . Since , it is possible to choose an appropriate constant such that and satisfy the conclusion of the theorem, which completes the proof. ∎
Proof of Lemma 2.2.
Given a -regular with vertices and smallest eigenvalue , we need to show that for every non-empty ,
Let denote the adjacency matrix of . For , let . Also, let be the -characteristic vector of . First note that . Since is a symmetric real matrix, it is diagonalizable by an orthonormal basis. Let be normalized eigenvectors of with corresponding eigenvalues , respectively. Since is a -regular graph, we have and . Let be the expansion of by eigenvectors. We have
Now observe that . In addition, . Therefore,
and the lemma follows. ∎
We now proceed to show robust stability for the Kneser graph for . The proof is a direct consequence of stability due to Friedgut  and a removal lemma for the Kneser graph due to Friedgut and Regev , which we state next.
Proposition 3.1 (Friedgut ).
Given , let be a sequence of integers satisfying . For all there exists and such that, for all , the following holds. If is an intersecting family of size at least , then there is such that .
Proposition 3.2 (Friedgut and Regev ).
Given , let be a sequence of integers satisfying . Moreover, let and . For all there exists and such that, for all , the following holds. Every family which spans at most non-intersecting pairs can be made intersecting by removing at most elements from .
Proof of Lemma 2.5.
It follows that for any family with and there exists an intersecting family obtained from by removing at most of its elements such that
In addition, Proposition 3.1 implies that for some , we have . Therefore,
which completes the proof. ∎
4 Proof of Proposition 2.6
We begin with the proof of a simple structural result for independent sets in graphs (Lemma 4.1). For a given graph , let denote the set of independent sets of of size exactly , and denote the set of all independent sets in .
Let be a graph on vertices, and be an arbitrary real number. In addition, let be integers. Then, for every independent set of size at least , there is a sequence of vertices and a sequence of subsets depending only on such that:
for all ,
for all .
Moreover, we have either
Fix an independent set of size at least . We need to define the required sequences and . Assume that we have already chosen elements and sets satisfying the conditions of our result. Observe that initially no element has been selected, and for convenience we set .
Consider an ordering of the vertices in which satisfies
for all and all . Such an ordering clearly exists, since one can repeatedly choose (and remove) the vertex with highest degree in the remaining graph. In this case we say that this is a max-ordering of the elements in .
Let be the smallest index such that the vertex in the max-ordering of is contained in . Such index must exist, since and . We define , and set .
If then we let . Note that, due to the max-ordering and the definition of , every vertex satisfies .This implies that the number of edges in satisfies . Otherwise, i.e. for the case , we let . Then,
Finally, observe that it follows from the definition of that we always have , which completes the proof. ∎
From this lemma we immediately deduce the following corollaries.
Let be a fixed ,-supersaturated graph on vertices with average degree , where . Let , and be an integer such that . Finally, set
Then, for every independent set , there exists a subset of size and a set , depending only on , of size at most such that . Further, we have . In particular, it follows that
Given , we apply Lemma 4.1 to obtain a sequence of vertices and sets , as stated. Now set and , and observe that and .
If for all , then . In other words, . On the other hand, if for some , then , since by assumption is -supersaturated. Altogether, it follows that , which completes the proof. ∎
Let and be graph on vertices which is -stab