Algorithmic Number On the Forehead Protocols Yielding Dense Ruzsa-Szemerédi Graphs and Hypergraphs

We describe algorithmic Number On the Forehead protocols that provide dense Ruzsa-Szemerédi graphs. One protocol leads to a simple and natural extension of the original construction of Ruzsa and Szemerédi. The graphs induced by this protocol have n vertices, Ω(n^2/log n) edges, and are decomposable into n^1+O(1/loglog n) induced matchings. Another protocol is an explicit (and slightly simpler) version of the construction of Alon, Moitra and Sudakov, producing graphs with similar properties. We also generalize the above protocols to more than three players, in order to construct dense uniform hypergraphs in which every edge lies in a positive small number of simplices.

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1 Introduction

For an integer and a positive real , let denote the maximum number so that any vertex graph with at least edges in which every edge is contained in a triangle, must contain an edge lying in at least

triangles. Erdős and Rothschild asked to determine or estimate

, see [5], [8], [9], [10]. Szemerédi observed that the triangle removal lemma (see [21]) implies that for every fixed , tends to infinity with , and Trotter and the first author noticed that for any there is a so that . A clever construction of Fox and Loh [13] shows that in fact for any fixed , . While this is still very far from the lower bound based on the triangle removal lemma and its improved quantitative version in [12], which provides a lower bound exponential in for any fixed , it does show that . Note that the constant is tight, as it is known that any -vertex graph with edges must contain an edge lying in at least triangles (see [15]).

The construction of Fox and Loh triggered another surprising result in the study of a closely related problem. The first author, Moitra and Sudakov [1] constructed -Ruzsa-Szemerédi graphs on vertices with and . A graph is an -Ruzsa-Szemerédi graph if its set of edges can be partitioned into pairwise disjoint induced matchings, each of size . These graphs were introduced in a paper by Ruzsa and Szemerédi [21]. They used these graphs, together with the regularity lemma of Szemerédi [23] to tackle the so called -problem dealing with the maximum possible number of edges of a -uniform hypergraph on vertices that contains no edges spanning at most vertices. Ruzsa-Szemerédi graphs have been studied extensively since, finding applications in Combinatorics, Complexity Theory and Information Theory. A natural line of research is to find dense graphs with relatively large . One such construction is given by Birk, Linial and Meshulam [4], with and . Meshulam conjectured that there are no -Ruzsa-Szemerédi graphs with both and . The construction from [1] disproved Meshulam’s conjecture in a strong form, vastly improving the one in [4].

The first aim of the present short paper is to describe these results in communication complexity terms by providing algorithmic Number-On-the-Forehead (NOF, for short) protocols that entail them. Ruzsa-Szemerédi graphs are closely related to the NOF model in communication complexity, as observed in [18]. They are related to the communication complexity of -dimensional permutations and sub-permutations (see details in the sequel). We observe here that communication protocols in the NOF model for -dimensional permutations also imply upper bounds on .

We give algorithmic NOF protocols that derive the constructions of dense Ruzsa-Szemerédi graphs from [1] and also the results of Fox and Loh [13]. This makes the constructions strongly explicit and also somewhat simpler. Another advantage of this approach is that it provides a clear link between these results and the original results of Ruzsa and Szemerédi [21].

The second aim of this paper is to extend the above mentioned results to uniform hypergraphs. To do so we extend the protocols to any number of players. Let denote the complete -uniform hypergraph (-graph, for short) on vertices. For an integer and a positive real , let denote the maximum number so that any vertex -graph with at least edges, in which every edge is contained in a copy of , must contain an edge lying in at least such copies. By the hypergraph removal lemma proved in [14] and independently in [20], [19], for any fixed positive , tends to infinity with . Indeed, for example, if is an -vertex -graph with at least edges, and each edge is contained in at least and at most copies of , then must contain at least pairwise edge-disjoint copies of . Hence at least that many edges have to be omitted from in order to destroy all copies of , and thus by the hypergraph removal lemma if is a constant then must contain at least copies of , implying that some edges are contained in such copies, contradiction.

Unlike the graph case, the maximum possible number of edges of an -vertex -graph with no copies of is not known. The determination of this number is an old problem posed by Turán [22], and Erdős offered a significant award for its solution, see [7]. By a general result proved in [16], the limit of the ratio

as tends to infinity exists. This is a positive number called the Turán density of . Let denote this number, which is conjectured to be for k=4. See [6] and its references for some of the work on this problem. Although is not known, we can prove the following.

Theorem 1.1

For any fixed there is some so that .

Note that by the results of [11] on supersaturated hyperghraphs if then any -graph on vertices with at least edges contains copies of . Therefore, for any such there is a constant so that , implying that the bound in Theorem 1.1 is tight.

Our protocols also imply an extension of the main result of [1]. That is, it entails a construction of nearly complete -graphs whose edges can be partitioned into a nearly linear number of induced subgraphs, each being a partial Steiner system. Recall that a -graph is a partial Steiner system if no two of its edges share common vertices. It is clear that any such graph on vertices cannot contain more than edges, and hence any -graph with at least edges cannot be partitioned into less than partial Steiner systems. The hypergraph removal lemma shows here, too, that in fact the number of such systems cannot be , that is, for any fixed positive , this number divided by must tend to infinity with . The following result shows, however, that this number can be smaller than for any positive .

Theorem 1.2

For every integer , there is an absolute constant so that for sufficiently large there is a -graph on vertices with at least

edges, whose edges can be decomposed into at most induced subgraphs, each being a partial Steiner system.

The rest of the paper contains the proofs of the above two theorems. The organization is as follows. Section 2 contains background on communication complexity and high-dimensional permutations, a recipe for proving Theorem 1.1 and Theorem 1.2 using communication protocols, and a simple application of this recipe to construct a graph on vertices and edges, decomposable into induced matchings. Section 3 contains the application of this recipe to prove Theorem 1.1 and Theorem 1.2. The details of the graphs and hypergraphs produced by this recipe, and the proof that it works correctly are given in Section 4. The final Section 5 contains a brief summary.

2 From communication to graphs and hypergraphs

2.1 Background and notation

General notation

We let . A -tuple is denoted either or in abbreviated form .

Communication complexity

We start with a few basic communication complexity notions. The definitions we give are a simplified version and adjusted to our needs. The interested reader can see [17] for a more comprehensive survey. In the NOF model players wish to compute a function . The players agree on a communication protocol . Then, an input is presented to the players so player sees all input except , we sometimes refer to this player as the -player. The players take turns to write messages on a blackboard according to the agreed protocol . Each message of each player may depend on the part of the input seen by this player, and except for the last player it can also depend on the messages written so far on the blackboard. The message written by the last player depends only on the part of the input he sees, and is independent of the content of the blackboard. One way to visualize this is as if the last player wrote a message first and then did not participate in the rest of the transaction. The value of the function can be computed by all players from the content of the board at the end of the protocol. The cost of a protocol, denoted , is the maximal number of bits written on the board, over all inputs, by the first players 111In the basic communication complexity definition all players can see each others messages, and the cost of the protocol depends also on the message of the last player. The version of communication complexity we gave here is from the one-sided model. Since we only need this version, we simplify our notations..

The string of bits written on the blackboard for a given input is called a transcript, denoted . We let for be the part of this transcript that is written by player . Let be a transcript, the subset of entries satisfying and , is called a cylinder intersection 222The usual definition of cylinder intersection is more general, what we defined here is referred to as a -monochromatic cylinder intersection. Since we are only interested in -monochromatic cylinder intersections we abbreviate the notation.. Note that a cylinder intersection is defined with respect to a function and a protocol for this function, we specify the function and protocol when it is necessary for a clear presentation and otherwise omit them.

We say that a subset of entries is symmetric if membership in does not depend on the order of the first entries. That is, is symmetric if if and only if for every permutation on .

High-dimensional permutations

A line in is a subset such that of the coordinates in are fixed, and the remaining coordinate takes all possible values. Following is a simple example with and :

In this example the first and third coordinates are fixed, and the second coordinate takes all possible values in . There is a distinct line for every choice of unconstrained coordinate , and a choice of values to fix the remaining coordinates. A line in is defined similarly. We say that the line is in the th dimension if the unconstrained coordinate is .

A -dimensional permutation is a function such that for every line in there is exactly one such that . A sub-permutation is a function such that every line in the th dimension contains a single , and every other line contains at most one .

For example, let be a group, define by if and only if . Then is a permutation. Let be a subset of , then the function defined similarly to , is a sub-permutation.

A weak permutation is a function such that every line contains at most one -entry, and a weak sub-permutation is defined similarly: it is an with such that every line contains at most one -entry.

Ruzsa-Szemerédi graphs and hypergraphs

As mentioned in the introduction, a graph is an -Ruzsa-Szemerédi graph if its set of edges can be partitioned into pairwise disjoint induced matchings, each of size . Such a graph obviously has edges. A challenge in constructing Ruzsa-Szemerédi graphs is to make the density of edges as large as possible while keeping the number of matchings relatively low. We are therefore less concerned with the size of each matching, and only worry about the number of matchings and the density of the edges.

There is a natural way to extend the notion of Ruzsa-Szemerédi graphs to hypergraphs, by considering Steiner systems . A Steiner system in a set , is a family of -element subsets of (called blocks) such that each -element subset of is contained in exactly one block. A partial Steiner system is defined similarly with the exception that each -element subset of is contained in at most one block.

For a natural number , and a -graph we are interested in partitioning into induced partial Steiner systems . Note that if is the set of vertices of a graph, then a partial Steiner system in is a matching. Thus, this definition extends the notion of a Ruzsa-Szemerédi graph.

2.2 A recipe

Given a function , a protocol for , and a transcript of the last player, denote

Next we describe a recipe for generating Ruzsa-Szemerédi graphs and hypergraphs, as well as upper bounds on , from NOF protocols.


Recipe 1
- from protocols to graphs and hypergraphs Choose a weak sub-permutation , for natural numbers , and . Construct a communication protocol for . Pick a transcript of the last player so that is symmetric, and let .

The following theorem describes the outcome when following Recipe 1.

Theorem 2.1

Let be a protocol found in the second step of Recipe 1, and let be the subset of inputs picked in the last step. Let , and , then

  1. There is an (explicitly defined) -graph on vertices whose edge density is , that is the union of induced partial Steiner systems .

  2. If , then for . Here, the construction of the -graph that gives the bound is also explicit, given explicit constructions of -graphs of density which contain no .

We defer the proof of Theorem 2.1 and the explicit definition of the graphs produced by Recipe 1 to Section 4. In the next section we give a simple example of how Theorem 2.1 can be applied, then in Section 3 we apply it to prove Theorems 1.1 and 1.2.

2.3 Applying Theorem 2.1 - an example

We apply Theorem 2.1 to prove:

Lemma 2.2

There is a graph on vertices with edge density that is the union of induced matchings.

Proof We follow the steps of Recipe 1:

Choosing the function

Let be natural numbers, denote , and define . Denote by the function satisfying if and only if (here addition is in ). It is not hard to verify that is a sub-permutation. Denote , then

Since , we have that as long as for some constant . We will later choose .

The protocol

Next we present a protocol for .


Protocol 1
A protocol for The -player computes , and writes the result on the board. The -player writes iff . The -player writes iff .

At the end, all players know the value of the function. Indeed, the value of the function is if the last two bits written on the board are both equal to , and otherwise.

The cost of the protocol

The cost of the protocol is , as the first two players send only verification bits.

The choice of

By the Chernoff-Hoeffding’s inequality (c.f., e.g., [2]), the quantity computed by the third player satisfies

Thus, with constant probability,

takes one of values. There is, therefore, a transcript for the third player such that . If we take we get . The fact that is symmetric is easy to verify. Lemma 2.2 now follows from Theorem 2.1, part 1.  

Note that we could improve the density of the graph in Lemma 2.2 to for any constant by taking for an appropriately chosen large constant . This seems to be the best one can get when using Protocol 1 though. In the next section we use a variant of this protocol in which the first two players participate more, in order to save communication bits of the last player. This will allow us to increase the density to near optimal.

3 Applying Theorem 2.1 to prove Theorems 1.1 and 1.2

3.1 The case

Choosing the function

The function we choose is , defined in Section 2.3. We later fix .

The protocol

For a natural number let be the graph with , where is even, and (later we take ). The players agree on a proper coloring of by colors, where is its maximum degree. Let , the players also agree on some partition of into intervals of length . The players choose that satisfy: the number of intervals in the partition is , and the number is in the middle of the interval containing it. As an example, the players can choose a partition which is a translation of the partition induced by . Let map a number in to the index of the interval containing it, according to . Given an input , the players then use the following protocol:


Protocol 2
A protocol for The -player writes on the board. The -player verifies that , and writes on the board iff this is the case. The -player verifies that , and writes on the board iff this is the case. If one of the last two bits are equal to , reject and finish. The -player writes on the board. The -player writes the value of .

Theorem 3.1

Protocol 2 is correct.

For the proof of correctness, we use the following two observations (used also in [1]):

Lemma 3.2 (Parallelogram law)

Let then:

Lemma 3.3 ([1])

For an even integer , the number of integral points contained in the ball of radius in is at most:

Proof [of Theorem 3.1] By Lemma 3.3, the maximum degree of is at most

The chromatic number of is therefore at most .

If then obviously the protocol reaches step 5. On the other hand, if the protocol reached step 5 then , , and , all lie in the same interval of length . Thus, by the Parallelogram law

Thus, is in a ball of radius around

. Every other vector

is in distance at most from . The color of in this ball is therefore unique. It follows that at step 6 the -player knows the value of and hence knows everything.  

The cost of the protocol

The number of bits used by the first two players is:

If we take , the cost of the protocol is therefore bounded by

The choice of

A transcript of the -player corresponds to a message . The size of is therefore equal to the number of pairs satisfying . Hoeffding’s inequality implies that

In particular, the probability that is at least since we chose the partition of the intervals so that lies in the middle of the interval containing it.

Take , and pick for , we have

Conclusion

When applying Theorem 2.1 the parameters that we get are:

  • ,

  • .

Taking , and observing that is symmetric, this proves the case of Theorems 1.1 and 1.2.

3.2 The case

Choosing the function

Let and define by if and only if . It is easy to verify that is a sub-permutation, and

The protocol

The protocol is a simple reduction to the case .


Protocol 3
A protocol for The first player writes on the board if and only if . If the last bit was equal to , the protocol ends with rejection. Players , and run Protocol 2 for with on input , and .

The correctness of the above protocol follows from the correctness of Protocol 2 and the fact that the equation holds if and only if . Note that the last equation cannot hold if does not belong to .

The cost of the protocol

Outside the reduction to Protocol 2, the players send only one more bit. The cost of the protocol thus satisfy , as before.

The choice of

We can choose, as in Section 3.1, the set for . By Hoeffding’s inequality, the size of is as long as . The only problem is that is not symmetric. To remedy that, just add to the protocol a test whether for every . These tests can all be carried out by the last player, so this adds only one more communication bit, which for simplicity we assume is the last bit. Now pick the transcript which imply that for all . The corresponding set is now symmetric, and as long as is a constant, Hoeffding’s inequality still implies that the size of is at least .

4 Proof of Theorem 2.1

We first rephrase Theorem 2.1 slightly.

Theorem 4.1

Let be a weak sub-permutation, and let be a symmetric cylinder intersection (w.r.t. ). Let , then

  1. There is an (explicitly defined) -graph on vertices whose edge density is , that is the union of induced partial Steiner systems .

  2. If , then for . Here, the construction of the -graph that gives the bound is explicit, given explicit constructions of -graphs of density which contain no .

Lemma 4.2

Theorem 4.1 implies Theorem 2.1.

Proof The difference between Theorem 4.1 and Theorem 2.1 lies in the different properties of the subset . In Theorem 2.1 is defined by

for some transcript of the last player. In Theorem 4.1 on the other hand, is a cylinder intersection, that is

for some transcript of all players.

This difference is easily bridged though. Let be a weak sub-permutation, a protocol for , a transcript of the last player, and a subset, found using Recipe 1. Let , and denote . For simplicity identify with .

Define by if and only if and . That is, is the message written on the board by the first players, according to protocol , on input .

It is not hard to verify that is a weak sub-permutation. We use the following protocol for , on input : the last player sends his message as in , then each of the other players verifies (using one bit of communication each) that his part in agrees with . Obviously is correct if and only if is correct. The subset

is a cylinder intersection with respect to and , and . Theorem 4.1 can now be applied to prove Theorem 2.1.  

In the rest of this section we prove Theorem 4.1. For simplicity we first prove it for the case of graphs () and then explain the necessary adjustments for the general case ().

4.1 The case

We prove the first conclusion of Theorem 4.1, concerning Ruzsa-Szemerédi graphs, in Section 4.1.1. The upper bound on is proved in Section 4.1.2. We use the following simple fact proved in [18].

Lemma 4.3 ([18])

Let be a function satisfying that every line in the third dimension contains at most a single , and let be a cylinder intersection (w.r.t ). Then, does not contain stars: triplets of the form where , and .

4.1.1 Ruzsa-Szemerédi graphs

The relation between Ruzsa-Szemerédi graphs and the communication complexity of -dimensional permutations was observed in [18]. The graphs constructed in [18] are bipartite though, and we need slightly different settings. Let be symmetric, define

Let be the graph with vertex set , where and , and edge set . We allow self loops in , and consider a collection of self loops as a matching. Note that when is a cylinder intersection with respect to a weak sub-permutation there is always at most one edge between a pair of vertices. The following lemma implies the first conclusion in Theorem 4.1.

Lemma 4.4

Let be a weak sub-permutation, and let be a symmetric cylinder intersection. Let be the subgraph of induced on . That is:

Then, the edges of can be partitioned into induced matchings.

Proof Partition the edge set as follows, for every let

This is a partition of since a sub-permutation, and therefore there is at most a single such that for every .

The fact that is an induced matching follows from Lemma 4.3. Assume in contradiction that is not an induced matching, then there is an edge for such that are in . We then get a star , contradicting Lemma 4.3. Note that the fact that is a sub-permutation also implies that and .  

4.1.2 An upper bound on

Consider the same graph as in the previous section. A basic observation is:

Lemma 4.5

Let be a function satisfying that every line in the third dimension contains at most a single , and let be a symmetric cylinder intersection (w.r.t ). Then, a triangle where and exists in if and only if .

Proof The fact that a triangle where and exists in for every follows immediately from the definition of . Assume in contradiction that there is also such a triangle in for . Then necessarily there are and such that . But then contains a star, in contradiction to Lemma 4.3.  

Lemma 4.6

Let be a weak sub-permutation, and let be a symmetric cylinder intersection satisfying . Then for .

Proof Consider the graph again. By lemma 4.5, and the fact that is a weak sub-permutation, an edge in appears in exactly one triangle with and . Therefore, if we take a bipartite subgraph inside , we will have every edge lie in exactly one triangle, which is optimal. But, the density of edges in is relatively small, since there are vertices and order of edges. To remedy this, we define a product function, aiming to increase the density of edges. The price we pay is that the number of triangles an edge can lie in increases.

Let be a natural number, define by if and only if . Let

It is not hard to verify that is a symmetric cylinder intersection with respect to . By Lemma 4.5 a triangle where and exists in if and only if . Thus, every edge of lies in at most triangles of this sort. To remove other kind of triangles let be a bipartite graph with density . Now define

Then every edge in lies in at least one triangle and at most triangles. The number of edges satisfy . The density of edges is thus

If we take this becomes

Recall that is a cylinder intersection of size . It therefore follows from the graph removal lemma (and the hypergraph removal lemma for larger ) - see Theorem 34 in [18] for details - that necessarily . The density is thus . Since every edge is in at most triangles, th