Every Steiner triple system contains an almost spanning d-ary hypertree

05/23/2021
by   Andrii Arman, et al.
Emory University
0

In this paper we make a partial progress on the following conjecture: for every μ>0 and large enough n, every Steiner triple system S on at least (1+μ)n vertices contains every hypertree T on n vertices. We prove that the conjecture holds if T is a perfect d-ary hypertree.

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1. Introduction

In this paper we study the following conjecture, raised by second author and Bradley Elliot [ER].

Conjecture 1.1.

Given there is , such that for any , any hypertree on vertices and any Steiner triple system on at least vertices, contains as subhypergraph.

Note that any hypertree can be embedded into any Steiner triple system , provided . The problem becomes more interesting if the size of the tree is larger. In [ARS] (see also [ER] Section 5) Conjecture 1.1 was verified for some special classes of hypertrees. In this paper we verify the conjecture for another class of hypertrees – perfect -ary hypertrees.

Definition 1.2.

A perfect -ary hypertree of height is a hypertree with , such that for all and for each and there are disjoint triplets with , .

In other words, is a perfect -ary hypertree if every non-leaf vertex has children (or a forward degree ). The main result of this paper is the following theorem.

Theorem 1.3.

For any real there is such that the following holds for all and any positive integer . If is a Steiner triple system with at least vertices and is a perfect -ary hypertree on at most vertices, then .

Remark 1.4.

One can consider the following extension of Theorem 1.3. In a perfect -ary hypertree label children of every vertex with numbers . Then every leaf can be identified with a sequence based on the way that leaf was reached from the root. We say that is an almost perfect -ary hypertree if is obtained from a perfect -ary hypertree by removing the smallest leafs in lexicographic order (for some integer )

With essentially same proof as of Theorem 1.3, for a fixed any sufficiently large Steiner triple system contains any almost perfect -ary hypertree with . For more extensions of Theorem 1.3, see Section 4.

2. Preliminaries

For positive integer let and for positive integers let . We write if . We write if is a union of disjoint sets and .

A hypertree is a connected, simple (linear) ‐uniform hypergraph in which every two vertices are joined by a unique path. A hyperstar of size centered at is a hypertree on vertex set with edge set . A Steiner triple system (STS) is a ‐uniform hypergraph in which every pair of vertices is contained in exactly one edge.

If is a hypergraph and , then (or when the context is clear) is the degree of a vertex in .

For we denote by the spanning subhypergraph of with

The proof of Theorem 1.3 relies on the application of an existence of almost perfect matching in an almost regular 3-uniform simple hypergraph. We will use two versions of such results. In the first version the degrees of a small proportion of vertices are allowed to deviate from the average degree. We will use Theorem 4.7.1 from [AS] (see [FR] and [PS] for earlier versions).

Theorem 2.1.

For any and there exists and such that the following holds. Let be a 3-uniform simple hypergraph on vertices and be such that

  • for all but at most vertices of the degree of

  • for all we have

Then contains a matching on at least vertices.

A second version is a result by Alon, Kim and Spencer [AKS], where under assumption that all degrees are concentrated near the average, a stronger conclusion may be drawn. We use a version of this result as stated in [KR]***We refer to Theorem 3 from that paper. There is a typo in the conclusion part of that theorem, where instead of there should be .

Theorem 2.2.

For any there exists such that the following holds. Let be a 3-uniform simple hypergraph on vertices and be such that for all . Then contains a matching on vertices.

Here the constant in -notation is depending on only an is independent of and .

In Lemma 3.3 we consider a random partition of the vertex set of Steiner triples system and heavily use the following version of Chernoff’s bound (this is Corollary 2.3 of Janson, Łuczak, Rucinski [JLR]).

Theorem 2.3.

Let

be a binomial random variable with the expectation

, then for

In particular for and

(1)

If is fixed and , then

(2)

3. Proof of Theorem 1.3

3.1. Proof Idea

Assume that is an STS on at least vertices. We will choose small constants .

Let be a perfect -ary tree on at most vertices with levels , and let . Let be a subhypertree of induced on . To simplify our notation we set and for all we set . Our goal is to find with such that , the subhypergraph induced by , contains a spanning copy of . In particular we would find such , level by level, first embedding , and then .

To start we consider a partition of with “random-like” properties (see Lemma 3.3 for the description of ) in the following way:

  • The first few levels of , constituting with the subset of leafs , will be embedded greedily into .

  • Then Lemma 3.1 and Lemma 3.2 will be used to find star forests in and for . The union of these star forests will establish embedding of almost all vertices of (see Claim 3.6).

  • Finally, the reservoir vertices will be used to complete the embedding (see Claim 3.7).

3.2. Auxiliary Lemmas

We start our proof with the following Lemma that allows us later to verify that contains almost perfect packing of size at most hyperstars centered at vertices of

Lemma 3.1.

For any positive real , there are and such that the following holds for all and all positive integers . Let be a 3-uniform simple hypergraph on vertices such that and

  • for all , and .

  • for all vertices we have

    and for all but at most vertices we have

  • for all .

Then contains a packing of hyperstars of size at most centered at vertices of that covers all but at most vertices.

Proof.

For given set and . With these parameters as an input, Theorem 2.1 yields and . Set and note that if Theorem 2.1 holds for some and then it also holds for smaller values of and larger values of . Therefore we may assume that is sufficiently small with respect to and .

Let that satisfies conditions (i)–(iii) be given. We start with constructing an auxiliary hypergraph that is obtained from by repeating the following splitting procedure for each vertex : split hyperedges incident to into disjoint groups, each of size , and then replace with new vertices and each hyperedge that belongs to group with a hyperedge .

First, we show that . Note that due to conditions (i)-(iii) the number of hyperedges

Provided is small enough and is large enough compared to we can guarantee . Finally, by construction of , so

Hypergraph satisfies assumptions of the Theorem 2.1 with parameters , , , and . Indeed, all of the vertices in still have degrees at most , and for all but at most vertices we have . Therefore, there is a matching in that omits at most vertices.

Now, matching in corresponds to a collection of hyperstars in with centers at vertices of and size of each is at most . Indeed, recall that during the construction of some vertices were replaced by vertices , hyperedges incident to were split into almost equal in size disjoint groups, and then each hyperedge in -th group was replaced with . Consequently, a matching in that covers some vertices gives a rise to a hyperstar centered at of size at most in .

Moreover since in omits at most vertices, the union of hyperstars also omits at most vertices. ∎

The following Lemma allows us later to verify that contains almost perfect packing of size at most hyperstars centered at vertices of for all .

Lemma 3.2.

For any positive real there is such that the following holds for all , and any positive integer . Let be a 3-uniform simple hypergraph on vertices such that and

  • for all , and .

  • for all and for all .

Then contains a packing of hyperstars of size at most centered at vertices of that covers all but at most vertices.

Here the constant in -notation depends on only.

Proof.

Proof is almost identical to the proof of Lemma 3.1. For a given let be the number guaranteed by Theorem 2.2 with as input.

Let that satisfies conditions (i),(ii) be given. We start with constructing an auxiliary hypergraph that is obtained from by splitting every vertex into new vertices that have degrees .

First, we will show that . Note that due to conditions (i) and (ii), we have

In particular,

As we have that and hence . Then by construction of , so as well.

Hypergraph satisfies assumptions of the Theorem 2.2 with parameters and . Therefore, there is a matching in that omits at most vertices.

Now, matching in corresponds to a collection of hyperstars in with centers at vertices of . Each hyperstar contains at most hyperedges and hyperstars cover all but at most vertices of , which finishes the proof. ∎

3.3. Formal Proof

We start by defining constants, proving some useful inequalities and proving Lemma 3.3.

Let be a Steiner triple system on vertices and let be the largest perfect -ary hypertree with at most vertices. Our goal is to show that .

We make few trivial observations. First, if and is perfect -ary hypertree with , then is just a hyperstar which clearly contains. Second, if , then can be found in greedily. Finally, if Theorem 1.3 holds for some value of , then Theorem 1.3 holds for larger values of . Hence we may assume without loss of generality that , and .

Constants. We will choose new constant independent of :

(3)

Let be a constant guaranteed by Lemma 3.1 with and as an input. We choose to be small enough, in particular we want

(4)

Properties of . Here we define the levels of and prove some useful inequalities. Recall that , denoted the levels of . For let be a subhypertree of induced on . To simplify our notation we also set and for all , and . Then we have for

(5)

Since , (5) implies and consequently

(6)

Finally, , where , so

(7)

Partition Lemma.

For a given STS with vertices our goal will be to find a partition of so that contains a copy of (and ), sets are the “candidates” for levels of and is a reservoir. Such a partition will be guaranteed by Lemma 3.3.

In the proof we will consider a random partition , where each vertex ends up in

with probability

and in with probability independently of other vertices.

To that end set

(8)

then by (7) and (4)

(9)

Now, for all define

(10)

Finally let . Then

and so

(11)

Hence .

Lemma 3.3.

Let , , , and be defined as above. Then for some and the following is true for any . If is a STS on vertices, then there is a partition of with the following properties:

  • for all .

  • for all and all

  • for all and all

  • for all ,

  • and contains a copy of a hypertree with as its last level. Moreover for all but at most vertices

    and for all vertices

Proof.

Recall that . Consider a random partition , where vertices are chosen into partition classes independently so that for while . For let be the event that the corresponding part of Lemma 3.3 fails. We will prove that for each .

Proof of Property (a). For all let be the event that

Then since and , Theorem 2.3 implies that

Since by (6), we infer that

Proof of Property (b). For all and let be the event

and be the event

Then and therefore for large enough . Moreover, Theorem 2.3 implies that

Finally, union bound yields

Proof of Property (c). Proof follows the lines of the proof of part (b). Since for and we have and . Hence we have

Proof of Property (d). Proof follows the lines of the proof of part (b), since for all we have and . Hence we have

Proof of Property (e)

We say that a set is typical if and contains a copy of . For a partition set for all .

Next we will show that the first statement of (e), namely that is typical, holds asymptotically almost surely.

Claim 3.4.
Proof.

Let be the event that and be the event that contains a copy of . Since and , Theorem 2.3 implies .

For let denote the event that , then . Indeed, if every vertex has degree at least is , then can be found in greedily, adding one hyperedge at a time.

Following the lines of proof of (d), we have , and

so by Theorem 2.3 for all we have Finally,

Therefore and hence . ∎

Now, for every typical set , we fix one copy of in .

We first show that there are not many vertices in that have low degree in .

Claim 3.5.

For any and any typical set all but at most vertices in satisfy

Proof of Claim.

For and there is a unique such that . Consequently, holds for any .

Let . Since for every there are at most edges with and

(12)

On the other hand let be the number of “bad” vertices , i.e., vertices with . Then we have

(13)

Comparing (12) and  (13) yields that . ∎

Let be the event that property (e) holds. Next we will show that

(14)

This implies that holds with probability .

Indeed, by Claim 3.4, and so

It remains to prove (14).

Denote by the space of all partitions of with for and , and for fixed let to be the space of all partitions of with probability function .

With this notation we need to show that for every typical .

Recall that and for all let

Note that for all

Then by (8)

(15)

Moreover, since for fixed the event was in the “initial” space independent of the outcome of random experiment for the remaining vertices , we infer that random variables are mutually independent.

Therefore for the rest of the proof we assume that typical with is fixed and all events and random variables are considered in the space .

For a typical define

(16)

Recall that since is typical we have

(17)

Then by Claim 3.5 with

(18)

Note that is independent of choice of and is fully determined by and .

Next we verify that certain events , , hold asymptotically almost surely and that . Let event be defined as

Then and , so