Approximations to m-coloured complete infinite hypergraphs

Approximations to $m$-coloured complete infinite hypergraphs

Abstract.

Given an edge colouring of a graph with a set of colours, we say that the graph is (exactly) -coloured if each of the colours is used. In 1999, Stacey and Weidl, partially resolving a conjecture of Erickson from 1994, showed that for a fixed natural number and for all sufficiently large , there is a -colouring of the complete graph on such that no complete infinite subgraph is exactly -coloured. In the light of this result, we consider the question of how close we can come to finding an exactly -coloured complete infinite subgraph. We show that for a natural number and any finite colouring of the edges of the complete graph on with or more colours, there is an exactly -coloured complete infinite subgraph for some satisfying ; this is best-possible up to the additive constant. We also obtain analogous results for this problem in the setting of -uniform hypergraphs. Along the way, we also prove a recent conjecture of the second author and investigate generalisations of this conjecture to -uniform hypergraphs.

2010 Mathematics Subject Classification:
Primary 05D10; Secondary 05C63, 05C65

1. Introduction

The classical problem of Ramsey theory is to find a large monochromatic structure in a larger coloured structure; for a host of results, see [4]. On the other hand, the objects of interest in anti-Ramsey theory are large ‘rainbow coloured’ or ‘totally multicoloured’ structures; see, for example, the paper of Erdős, Simonovits and Sós [2]. Between these two ends of the spectrum, one could consider the question of finding structures which are coloured with exactly different colours: this was first done by Erickson [3] and this is the line of enquiry that we pursue here.

Our notation is standard. Thus, following Erdős, for a set , we write for the family of all subsets of of cardinality ; equivalently, is the complete -uniform hypergraph on the vertex set . We write for , the set of the first natural numbers. We denote a surjective map from a set to another set by . By a colouring of a hypergraph, we mean a colouring of the edges of the hypergraph unless we specify otherwise.

Let be a surjective -colouring of the edges of the complete -uniform hypergraph on the natural numbers. We say that a subset is (exactly) -coloured if , the set of values attained by on the edges induced by , has size exactly . Let , or in short, denote the size of the set ; in other words, every set is -coloured. In this paper, we shall study for fixed and large , the set of values for which there exists an infinite -coloured set with respect to a -colouring . Let us mention as an aside that it is also interesting to study what happens when we wish to find finite -coloured sets, or allow colourings which use infinitely many colours; we refer the reader to [5] for results of this flavour. With our goal of finding infinite -coloured sets in mind, let us define, for a -colouring , the set

Clearly, as is surjective, and Ramsey’s Theorem tells us that . Erickson [3] noted that a fairly straightforward application of Ramsey’s Theorem enables one to show that for any -colouring of with . He also conjectured that with the exception of , no other elements are guaranteed to be in (even in the case of graphs) and that if , then there is a -colouring of such that . Stacey and Weidl [7], partially resolving this conjecture, showed using a probabilistic construction that there is a constant such that if , then there is a -colouring of such that .

Since an exactly -coloured complete infinite subhypergraph is not guaranteed to exist, we are naturally led to the question of whether we can find a complete infinite subhypergraph that is exactly -coloured for some close to . In this paper, we establish the following result.

Theorem 1.1.

Fix a positive integer . For any -colouring and any natural number , there exists an such that

where .

Theorem 1.1 is tight up to the term. To see this, let for some . We consider the ‘small-rainbow colouring’ which colours all the edges induced by with distinct colours and all the remaining edges with the one colour that has not been used so far. In this case, we see that . Now let for some natural number such that . It is not difficult to check that for each ; also, it is clear that .

In the case of graphs where , Theorem 1.1 tells us that for any finite colouring of the edges of the complete graph on with or more colours, there is an exactly -coloured complete infinite subgraph for some satisfying ; a careful analysis of the proof of Theorem 1.1 in this case allows us to replace the term with an explicit constant, .

We know from Theorem 1.1 that cannot contain very large gaps. Another natural question we are led to ask is if there are any sets, and in particular, intervals that is guaranteed to intersect. Making this more precise, the second author conjectured, see [6], that the small-rainbow colouring described above is extremal for graphs in the following sense.

Conjecture 1.2.

Let be a -colouring of the complete graph on and suppose is a natural number such that . Then .

In this paper, we shall prove this conjecture. There are two natural generalisations of this conjecture to -uniform hypergraphs which are equivalent to Conjecture 1.2 in the case of graphs.

The first comes from considering small-rainbow colourings; indeed we can ask whether when , where is the interval .

The second comes from considering a different family of colourings which we call ‘small-set colourings’. Let and consider the surjective -colouring of defined by . Note that in this case, . Consequently, we can ask whether when , where is the interval .

Note that both these questions are identical when . Indeed, , so .

We shall demonstrate that the correct generalisation is the former. We shall first prove that the answer to the first question is in the affirmative, provided is sufficiently large.

Theorem 1.3.

For every , there exists a natural number such that for any natural number and any -colouring with , .

Using a result of Baranyai [1] on factorisations of uniform hypergraphs, we shall exhibit an infinite family of colourings that answer the second question negatively for every .

Theorem 1.4.

For every , there exist infinitely many values of for which there exists a -colouring with such that .

The rest of this paper is organised as follows. In the next section, we shall prove Theorems 1.11.3 and 1.4 and deduce Conjecture 1.2 from the proof of Theorem 1.3. We then conclude by mentioning some open problems.

2. Proofs of the main results

We start with the following lemma which we shall later use to prove both Theorems 1.1 and 1.3.

Lemma 2.5.

Let be an element of . Then there exists a natural number such that

  1. , and

  2. .

Futhermore, if

for some and , then

Proof.

We start by establishing the following claim.

Claim 2.6.

There is an infinite -coloured set with a finite subset such that

  1. the colour of every edge of is determined by its intersection with , i.e., if , then , and

  2. for all .

Proof.

To see this, let be an infinite -coloured set. For each colour , pick an edge in of colour and let be the set of vertices incident to these edges. So is a finite -coloured set. Let be an enumeration of the subsets of of size at most . Note that this is the complete list of possible intersections of an edge with . We now define a descending sequence of infinite sets as follows. Let . Having defined the infinite set , we induce a colouring of the -tuples of , by giving the colour of the edge . By Ramsey’s Theorem, there is an infinite monochromatic subset with respect to this induced colouring, so the edges of whose intersection with is have the same colour.

Hence, is an infinite -coloured set satisfying property (1). Now, if we have a vertex such that , we delete from . We repeat this until we are left with an -coloured set satisfying (1) and (2). ∎

Let and be as guaranteed by Claim 2.6. Note that is nonempty since . We shall prove the lemma with . From the structure of and , we note that . That

is a consequence of the following claim.

Claim 2.7.

There exist infinite sets such that and .

Proof.

Let for any . We know from Claim 2.6 that . We shall now prove that ; that is, the number of colours lost by removing from is at most . Since the colour of an edge is determined by its intersection with , the number of colours lost is at most the numbers of subsets of containing of size at most , which is precisely .

Next, we shall prove that there is a subset such that . Let and let

be the set of colours lost by removing from ; since for all , it follows that . For each colour , pick an edge of colour , and let ; in particular, we take , where is the colour corresponding to an empty intersection with . Since every edge of colour contains , we double count the number of times a colour is counted in the sum to obtain

so there exists an such that ; the claim follows by taking . ∎

We finish the proof of the lemma by establishing the following claim.

Claim 2.8.

If we can write for some and , then

Proof.

As in the proof of Claim 2.7, for each colour , pick an edge of colour , and let ; in particular, let . We know from Claim 2.6 that edges of of distinct colours cannot have the same intersection with . Consequently, all the are distinct subsets of , each of size at most . Hence,

Arguing as in the proof of Claim 2.7, we conclude that there exists a vertex such that the number of colours lost by removing from is at most . Therefore,

so it follows that

The lemma now follows from Claims 2.62.7 and 2.8. ∎

Having established Lemma 2.5, it is easy to deduce both Theorem 1.1 and 1.3 from the lemma.

Proof of Theorem 1.1.

Let . We may assume that since otherwise and there is nothing to prove. Also, if , then the result follows easily by taking so we may assume that . Let be the smallest element of greater than . Applying Lemma 2.5 to , we find an such that and

for some natural number . Now if , then

so it follows that . If on the other hand, then using the fact that

it follows once again that . ∎

Proof of Theorem 1.3.

If , we are done since . So suppose that . Let be the smallest element of such that ; hence, . Now, since , there exists by Lemma 2.5, a natural number such that

To prove the theorem, it is sufficient to show that . We know from Lemma 2.5 that . If is sufficiently large, we must have .

If , then

since and .

We now deal with the case . First, we write . Since and , we see that . By Lemma 2.5, it follows that

Since and , the result follows. ∎

A careful inspection of the proof of Theorem 1.3 shows that when , the statement holds for all . We hence obtain a proof of Conjecture 1.2. By constructing a sequence of highly structured subgraphs, the second author [6] proved that for any -colouring with for some natural number , ; our proof of Conjecture 1.2 gives a short proof of this lower bound. Theorem 1.3 also yields a generalisation of this lower bound for -uniform hypergraphs, albeit with a constant additive error term (which depends on ).

We now turn to the proof of Theorem 1.4. We will need a result of Baranyai’s [1] which states that the set of edges of the complete -uniform hypergraph on vertices can be partitioned into perfect matchings when .

Proof of Theorem 1.4.

We shall show that if is sufficiently large and , then there is a surjective -colouring of with and . We shall define a colouring of such that the colour of an edge is determined by its intersection with a set of size , say . Let be the family of all subsets of of size at most . For , we denote the colour assigned to all the edges such that by .

To define our colouring, we shall construct a partition with . Then for every , we set to be equal to . Finally, we take the colours for to all be distinct and different from . Hence, the number of colours used is . It remains to construct this partition of .

Since , by Baranyai’s theorem there exists an ordering

of the subsets of of size such that for all , the family

is a perfect matching. Let , where

our colouring is well defined because for all sufficiently large . Observe that

We shall show that the second largest element of is at most . Note that any with cannot contain . As before, let be the set of colours lost by removing from , i.e.,

We shall complete the proof by showing that for all .

Note that our construction ensures that for all . Now, observe that

so for all . It is then easily verified using Pascal’s identity that when and is sufficiently large,

the last inequality above is deduced by comparing the coefficients of the polynomials in the inequality.

When , it is easy to check that , so is divisible by . Consequently, in this case, for . Hence,

This completes the proof. ∎

3. Conclusion

We conclude by mentioning two open problems. We proved that for any -colouring and every sufficiently large natural number , provided . A careful analysis of our proof shows that the result holds when ; we chose not to give details to keep the presentation simple. However, we suspect that the result should hold as long as but a proof eludes us.

To state the next problem, let us define

A consequence of Theorem 1.3 is that . Turning to the question of upper bounds for , the small-rainbow colouring shows that the lower bound that we get from Theorem 1.3 is tight infinitely often, i.e., when is of the form for some . When is not of this form, there are two obvious ways of generalising the small-rainbow colouring: we could replace the rainbow coloured clique in our construction either with a disjoint union of cliques or with a clique along with a pendant vertex attached to some subset of the vertices of the clique. However, both these obvious generalisations of the small-rainbow colouring fail to give us good upper bounds for for a general . The second author proved [6] using rainbow colourings of complete bipartite graphs that

for almost all natural numbers and some absolute constant . The same construction can be extended to show that for almost all natural numbers . It would be very interesting to decide if, in fact, for all .

Acknowledgements

Some of the research in this paper was carried out while the authors were visitors at Microsoft Research, Redmond. We are grateful to Yuval Peres and the other members of the Theory Group at Microsoft Research for their hospitality.

We would also like to thank Tomas Juškevičius for helpful discussions.

References

  1. Zs. Baranyai, On the factorization of the complete uniform hypergraph, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Colloq. Math. Soc. János Bolyai, vol. 10, North-Holland, Amsterdam, 1975, pp. 91–108.
  2. P. Erdős, M. Simonovits, and V. T. Sós, Anti-Ramsey theorems, Infinite and finite sets (Colloq., Keszthely, 1973; dedicated to P. Erdős on his 60th birthday), Colloq. Math. Soc. János Bolyai, vol. 10, North-Holland, Amsterdam, 1975, pp. 633–643.
  3. M. Erickson, A conjecture concerning Ramsey’s theorem, Discrete Math. 126 (1994), 395–398.
  4. R. L. Graham, B. L. Rothschild, and J. H. Spencer, Ramsey theory, 2nd ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1990.
  5. T. Kittipassorn and B. Narayanan, A canonical Ramsey theorem for exactly m-coloured complete subgraphs, Combin. Probab. Comput., To appear.
  6. B. Narayanan, Exactly -coloured complete infinite subgraphs, J. Combin. Theory Ser. B, To appear.
  7. A. Stacey and P. Weidl, The existence of exactly -coloured complete subgraphs, J. Combin. Theory Ser. B 75 (1999), 1–18.
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