Ramsey properties of randomly perturbed dense graphs

Ramsey properties of randomly perturbed dense graphs

Abstract.

We investigate Ramsey properties of a random graph model in which random edges are added to a given dense graph. Specifically, we determine lower and upper bounds on the function that ensures that for any dense graph  a.a.s. every 2-colouring of the edges of admits a monochromatic copy of the complete graph . These bounds are asymptotically sharp for the cases when is odd and almost sharp when is even. Our proofs utilise recent results on the threshold for asymmetric Ramsey properties in and the method of dependent random choice.

A large part of this work forms the Bachelor thesis of the author which was supervised by Mathias Schacht (schacht@math.uni-hamburg.de) and handed in at the Department of Mathematics at the University of Hamburg in March 2018. While we prepared this manuscript, Das and Treglown [8] independently obtained similar and more general results (see Section 1.4)

.

1. Introduction

1.1. Random graphs and randomly perturbed dense graphs

For and we denote by the binomial random graph on  vertices where every edge is present with probability independently of all other choices. As usual, we say that an event happens asymptotically almost surely (a.a.s.) if it holds with probability tending to as . Given a graph property , it has been a key question to find a threshold function, a function ensuring that a.a.s. satisfies when and a.a.s. does not satisfy when

Bohman, Frieze and Martin [5] considered a model that combines deterministic graphs and random graphs: In that model of randomly perturbed graphs one starts with an arbitrary dense graph and adds edges in a random manner. More precisely, given , we say that a graph is -dense if Furthermore, we say that ensures a property if

where the minimum is taken over all -dense graphs on the same vertex set as . For a fixed , we say that a function is a threshold for (in the context of randomly perturbed dense graphs) if for and for Throughout, we will assume that is some fixed and small constant.

Stricly speaking, working in this model requires to consider sequences of -dense graphs . However, for a better presentation, we suppress the sequences and similarly we simply write for .

Recently, several thresholds for this model have been studied in [4, 19, 7, 18, 1, 17, 2, 3, 6, 12, 15, 22, 9]. Most of the analysis centered around ensuring spanning structures such as trees or (powers of) cycles. Krivelevich, Sudakov and Tetali [19] already investigated Ramsey properties of this model (see Section 1.3 below). We continue this line of research (see Section 1.4).

1.2. Ramsey properties of random graphs

For graphs , we denote by the Ramsey-type statement that every colouring of with colours yields a monochromatic copy of in colour for some . In the symmetric case when , we simply write and if additionally , then we write . Using this notation, Ramsey’s theorem states that for all there exists some such that .

Rödl and Ruciński established the threshold for the property which for every fixed graph . For a graph we define

and we let denote the 2-density, defined by . The following is a slightly simplified version of the result mentioned above:

Theorem 1 (Rödl and Ruciński [23, 24]).

Let be an integer and let be a graph that is not a forest. Then there exist real constants , such that

Recently, one focus of research in this area is to establish thresholds for asymmetric Ramsey properties. Interestingly enough, some of the recent discoveries will play a key role in the proofs of our results and will be introduced in Section 3.

1.3. Ramsey properties of randomly perturbed dense graphs

Concerning the newer model of randomly perturbed graphs, a first reasonable question to address is whether the Rödl-Ruciński threshold can be improved at all. In fact this cannot be achieved for (i.e. more than 2 colours) and small : In case of (for example, when and is a clique) there exists an -free, -dense graph . Then we can assign one colour to all edges from without admitting a monochromatic copy of in that colour. We still have at least two unused colours left to cope with the edges of the random graph, so we will be able to colour the remaining edges without admitting a monochromatic copy of , unless we have . By Theorem 1 a threshold for is also a threshold for and thereby for by the above consideration. Hence, we will focus on the case only. We first recall the Ramsey result from [19].

Theorem 2 (Krivelevich, Sudakov and Tetali [19]).
  1. If , then for any , any integer and any -dense -vertex graph we a.a.s have

  2. If , then for any constant and for every there exists a -dense -vertex graph such that we a.a.s. have

Note that in particular this shows that is a threshold function for .

1.4. Our results

As mentioned above, our proofs are based on asymmetric Ramsey results and these involve the asymmetric 2-density . For two graphs  and , both having at least one edge, let

and let

Then our first result reads as follows.

Theorem 3.

Let be a real constant and let be an integer.

  1. Let be a constant and . For any -dense -vertex graph we a.a.s. have

  2. Let be odd. Then there exists a real constant such that the following holds for . For any -dense -vertex graph we a.a.s. have

We complement Theorem 3 by the following lower bound for the threshold.

Theorem 4.

Let be an integer and let be a positive constant. Then there is a real constant and an -vertex graph with such that for we a.a.s. have

Note that Theorem 4 shows that Theorem 32 is asymptotically optimal for odd while for even Theorem 31 leaves a ‘gap’ of an arbitrarily small in the exponent. Independently of our work, Das and Treglown [8] proved a more general result which closes these gaps. Finally, the following theorem covers the remaining case .

Theorem 5.
  1. Let and . Then for any graph on vertices with at least edges and any we a.a.s. have

  2. For there exists a graph on vertices with at least edges such that for we a.a.s. have

Again, the result of Das and Treglown [8] improves the upper bound in Theorem 51, while Theorem 52 improves their lower bound. Thus, both results together establish as a threshold for .

2. Preliminaries & Notation

In this short section we introduce further notations and a few basic results that we will use repeatedly. For a graph and a subset , by we denote the subgraph of induced by the vertex set . Furthermore, we will sometimes write instead of .

By we denote the density of a graph which is defined as

The following well-known result establishes the threshold for containing a fixed subgraph.

Theorem 6 ([10]).

Let be an integer. We have

In order to get exponential bounds on the probability of non-existence of a graph we will use the following variant of Janson’s inequality (see [13], [14]).

Theorem 7 ([13], [14]).

For any there exists some constant such that for any and all the probability that is -free is at most

3. Proofs

3.1. Proof of Theorem 4

We start off by showing the optimality of our result for as this proof is easy and still already illustrates the relation between Ramsey properties of the considered model and (asymmetric) Ramsey properties of pure random graphs. The following -statement for an asymmetric Ramsey property is crucial. Note that the result of Marciniszyn, Skokan, Spöhel and Steger [21] is more general as it addresses an arbitrary number of colours and cliques.

Theorem 8 ([21]).

Let be integers. Then there exists a real constant such that for we a.a.s. have

Proof of Theorem 4.

Let be the complete bipartite graph . Then, for sufficiently large we have

as required. We obtain from Theorem 8.

We get

where the equality follows from Theorem 8 and the inequality follows from the following deterministic argument.

Suppose that for an -vertex graph and let be a red-blue-colouring of that does not admit a red copy of or a blue copy of . We extend to a colouring of by assigning the colour red to all remaining edges. The resulting colouring does not admit a blue copy of , since does not. Futhermore, it does not yield a red copy of , because any such copy would need to have at least vertices in one of the partition classes of the bipartite graph , contradicting the choice of . ∎

3.2. Proof of Theorem 3

The proof of our result for cliques of even size works as follows: We first show that in order to satisfy it suffices to have for all ’almost linear’ subsets (those of size at least ). In the second step we bound the probability of the above event by means of an asymmetric Ramsey result that was established recently.

For odd we can improve the technique by seeking for in ’linear subsets’ and in the ’almost linear subsets’. Since the latter property generally requires a smaller to be ensured, the first will be the limiting factor, although we ask for it in slightly bigger subsets. By means of this little trick, the reducing the size of our subsets in the second condition will no longer play a role and we get the asymptotically tight 1-statement. A key element for the first step (of both proofs) is the following lemma given by Fox and Sudakov in their article on the method of dependent random choice [11]:

Lemma 9 ([11]).

Let , , , and be positive integers. Let be a graph with and average degree . If there is a positive integer t such that

(1)

then contains a subset of at least vertices such that every vertices in have at least common neighbours.

Corollary 10.

For any , and there is a constant and such that the following holds for all integers . For every -dense -vertex graph there is a subset with such that every vertices in have at least common neighbours.

Proof.

The following is a straightforward application of the Lemma we just introduced. As usual, for the sake of a less baroque presentation, we do not round our parameters and instead assume they are integers. For and let be sufficiently large such that . For any integer and a given graph satisfying the above properties we then get

which verifies (1) for and . Hence, the corollary follows from Lemma 9. ∎

The following lemma completes step 1.

Lemma 11.

Let , be real constants and let . Then there exist a real constant and such that the following holds for all integers and all -dense -vertex graphs .

(2)
Proof.

Let and be given by Corollary 10. Let be a graph satisfying and the other assumptions from above. Suppose for contradiction that is a red-blue-colouring of without a monochromatic copy of . Without loss of generality we may assume and let be the subgraph on that contains only the red edges. By Corollary 10 there exists with such that every vertices from have at least common neighbours in . It follows from our assumption that we have Since does not contain a blue copy of , there must be a copy of in . Let be the set of vertices inducing this copy. Owing to and the choice of , the common neighbourhood of in has size at least . Now the second assumption implies and by the argument given above we find a copy of in . Let be the set of vertices forming this copy; then induces a copy of in , yielding a contradiction. ∎

As indicated above, in step 2 we want to bound the probability of the event forming the hypothesis in (2) using asymmetric Ramsey results.

Not too long ago, Kohayakawa, Schacht and Spöhel [16] proved an asymmetric 1-statement for two well-behaved graphs and . In particular, for cliques their upper bound asymptotically coincides with the lower bound we met in Theorem 8. However, for our purposes their main result does not help much because we need the specific exponential form of the error probability. This is because for our arguments the error probability still needs to converge to 0 when multiplied with the number of subsets of the vertex set, i.e. .

Instead we want to apply Lemma 23 from [16] which (in the original form) contains some paper-specific notation that we do not need. Hence, we state a version that is well adapted to our setting and provide a short deduction in the following lines. However, we do not go into any details of [16]:

For our purposes we take and . It is easy to check that  is strictly balanced with respect to (see p.3 from that paper for the definition). Then Lemma 13 from that article assures that and satisfy the hypothesis of Lemma 23.

In the lemma you find a graph parameter (where, of course,  is a graph). Since we do not intend to use its explicit form anywhere, we refer to [16] (Definiton 11) in case the reader wishes to have a look at the definiton.

Lemma 12 ([16]).

Let be integers. Then there exist real constants , and such that for any integer and any satisfying

we have

Additionally, we need to know that there exists a that lies between the bounds given in Lemma 12. The following lemma ensures that and it follows from Lemma 13 (in [16]), once more using that is strictly balanced with respect to .

Lemma 13 ([16]).

For integers we have

With this in hand, we will be able to complete step 2. By the union bound, the probability that the hypothesis of (2) fails can now be bounded by a term of the form

(or without the in the ’improved setting’) and it will turn out that this term converges to for our range of . Note that this is not surprising as is of the same order as the expected number of copies of which should outdo  (see parts 2 and 3 of Lemma 14). We hope that this already outlines the proofs reasonably well. Still, on the next pages we present the proofs for both cases in great detail. In order to make the proofs less technical, we extract some calculations revolving around the asymmetric 2-density into another lemma.

Lemma 14.

Let be an integer and let , .

  1. For cliques the asymmetric 2-density has the form

    where is another integer.

  2. If and , then we have

  3. If is odd, and , then we have

  4. Finally, for odd we have

Proof.
  1. This follows from

    and

  2. Using 1 we get

  3. The first inequality in the statement is trivial and the second can be verified by

  4. This can be seen easily after applying 1 to both sides.

We start with the general case where is an arbitrary integer and is given.

Proof of Theorem 31.

Throughout this proof let and . By monotonicity we may assume and . We start by applying our preparatory lemmas:

  • Lemma 12 yields constants , and such that

    (3)

    for all integers and any with

  • Lemma 11 yields and such that (2) holds for graphs with .

Let satisfy

and additionally

for all integers the latter is possible due to the assumption we made in the beginning of the proof. Now let an integer be given. By monotonicity we may assume

Then for any with we may apply (3) to (which we may identify with the random graph ), since we have

and

(3) yields

(4)

Owing to , the following statement implies (2):

Therefore, we have

where the last inequality uses Lemma 142. This proves the theorem since the last term converges to when goes to infinity. ∎

Now as to the more specific case where is odd:

Proof of Theorem 32..

Throughout the proof let , and By Lemma 144 there exists a such that and we let . Again, we start by applying our preparatory lemmas:

  • Lemma 12 yields constants , and such that

    (5)

    for all integers and any with

  • Lemma 12 yields constants , and such that

    (6)

    for all integers and any satisfying

  • Lemma 11 yields and such that (2) holds for graphs with . We may assume .

Now let

and let satisfy and additionally

for all integers , which is possible by Lemma 13. Now let an integer be given. By monotonicity we may assume

  • For any with we may apply (5) to (which we may identify with ), as we have and

    (5) shows

    (7)
  • Analogously, if , for any with we may apply (6) to since we have and

    where we used the definitions of and . We get

    (8)

    where with being the constant yielded by Theorem 7. This follows from (6) for and from Theorem 7 for . Note that in the latter case , and thus is just containing a copy of . Therefore, the asymmetric Ramsey result is not applicable in that case and we have to go for Janson’s.

Using (2) we get

where the last inequality follows from Lemma 143, which is applicable due to . This proves the theorem because the last term goes to as  approaches infinity. ∎

3.3. Brief discussion of and the proof of Theorem 5

Considerations are of a different nature for the clique sizes and . In these cases comes down to containing a copy of . Proceeding as in Theorem 4, we obtain a lower bound of (see Theorem 6). We also obtain an upper bound based on Lemma 11; however, we need  to be of order

as we can only apply this technique if the expected number of copies of is linear. This leaves us with a significant gap between the bounds. For the threshold coincides with the lower bound (see Theorem 2), whereas for we will improve the lower bound so that it matches the order of the upper bound. Let us start with a brief proof of the upper bound; the method is the same as in the proof of Theorem 3 where we went into great detail.

Proof of Theorem 51.

Following the lines of the proof of Theorem 31 and applying Theorem 7 yields

for a sufficiently large and an appropriate constant . Clearly, the last term converges to as desired. ∎

We now turn our attention to the lower bound and start by introducing our key lemma which we will prove later. We write if every 2-colouring of admits a monochromatic copy of