Algorithmic Number On the Forehead Protocols Yielding Dense Ruzsa-Szemerédi Graphs and Hypergraphs
aa r X i v : . [ c s . CC ] J a n Algorithmic Number On the Forehead Protocols YieldingDense Ruzsa-Szemer´edi Graphs and Hypergraphs
Noga Alon ∗ Princeton UniversityPrinceton, NJ 08544, USAand Tel Aviv UniversityTel Aviv 69978, Israel [email protected]
Adi ShraibmanThe Academic College of Tel-Aviv-YaffoTel-Aviv, Israel [email protected]
Abstract
We describe algorithmic Number On the Forehead protocols that provide dense Ruzsa-Szemer´edigraphs. One protocol leads to a simple and natural extension of the original construction of Ruzsaand Szemer´edi. The graphs induced by this protocol have n vertices, Ω( n / log n ) edges, andare decomposable into n O (1 / log log n ) induced matchings. Another protocol is an explicit (andslightly simpler) version of the construction of [1], producing graphs with similar properties. Wealso generalize the above protocols to more than three players, in order to construct dense uniformhypergraphs in which every edge lies in a positive small number of simplices. For an integer n and a positive real c , let h ( n, c ) denote the maximum number so that any n vertexgraph with at least cn edges in which every edge is contained in a triangle, must contain an edgelying in at least h ( n, c ) triangles. Erd˝os and Rothschild asked to determine or estimate h ( n, c ), see[5], [8], [9], [10]. Szemer´edi observed that the triangle removal lemma (see [21]) implies that forevery fixed c > h ( n, c ) tends to infinity with n , and Trotter and the first author noticed that forany c < / c ′ so that h ( n, c ) < c ′ √ n . A clever construction of Fox and Loh [13] showsthat in fact for any fixed c < / h ( n, c ) ≤ n O (1 / log log n ) . While this is still very far from the lower ∗ Research supported in part by NSF grant DMS-1855464, ISF grant 281/17, BSF grant 2018267 and the SimonsFoundation. ound based on the triangle removal lemma and its improved quantitative version in [12], whichprovides a lower bound exponential in log ∗ n for any fixed c >
0, it does show that h ( n, c ) = n o (1) .Note that the constant 1 / n -vertex graph with ⌊ n / ⌋ + 1 edgesmust contain an edge lying in at least n/ r, t )-Ruzsa-Szemer´edigraphs on n vertices with r = n − o (1) and rt = (1 − o (1)) (cid:0) n (cid:1) . A graph is an ( r, t )-Ruzsa-Szemer´edigraph if its set of edges can be partitioned into t pairwise disjoint induced matchings, each of size r . These graphs were introduced in a paper by Ruzsa and Szemer´edi [21]. They used these graphs,together with the regularity lemma of Szemer´edi [23] to tackle the so called (6 , n vertices that containsno 3 edges spanning at most 6 vertices. Ruzsa-Szemer´edi graphs have been studied extensively since,finding applications in Combinatorics, Complexity Theory and Information Theory. A natural lineof research is to find dense graphs with relatively large r . One such construction is given byBirk, Linial and Meshulam [4], with r = (log n ) Ω(log log n/ (log log log n ) ) and t = Ω( n /r ). Meshulamconjectured that there are no ( r, t )-Ruzsa-Szemer´edi graphs with both rt = Θ( (cid:0) n (cid:1) ) and r ≥ n Ω(1) .The construction from [1] disproved Meshulam’s conjecture in a strong form, vastly improving theone in [4].The first aim of the present short paper is to describe these results in communication complexityterms by providing algorithmic Number-On-the-Forehead (NOF, for short) protocols that entailthem. Ruzsa-Szemer´edi graphs are closely related to the NOF model in communication complexity,as observed in [18]. They are related to the communication complexity of 2-dimensional permu-tations and sub-permutations (see details in the sequel). We observe here that communicationprotocols in the NOF model for 2-dimensional permutations also imply upper bounds on h ( n, c ).We give algorithmic NOF protocols that derive the constructions of dense Ruzsa-Szemer´edigraphs from [1] and also the results of Fox and Loh [13]. This makes the constructions stronglyexplicit and also somewhat simpler. Another advantage of this approach is that it provides a clearlink between these results and the original results of Ruzsa and Szemer´edi [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 k > K k = K ( k − k denote thecomplete ( k − k − k vertices. For an integer n anda positive real c , let h k − ( n, c ) denote the maximum number so that any n vertex ( k − cn k − edges, in which every edge is contained in a copy of K k , must contain an edgelying in at least h k − ( n, c ) such copies. By the hypergraph removal lemma proved in [14] andindependently in [20], [19], for any fixed positive c , h k − ( n, c ) tends to infinity with n . Indeed,for example, if G is an n -vertex 3-graph with at least cn edges, and each edge is contained in atleast 1 and at most h = h ( n, c ) copies of K = K , then G must contain at least cn h pairwiseedge-disjoint copies of K . Hence at least that many edges have to be omitted from G in order to estroy all copies of K , and thus by the hypergraph removal lemma if h is a constant then G mustcontain at least Ω( n ) copies of K , implying that some edges are contained in Ω( n ) such copies,contradiction.Unlike the graph case, the maximum possible number ex k − ( n, K k ) of edges of an n -vertex( k − K k is not known. The determination of this number is an oldproblem posed by Tur´an [22], and Erd˝os offered a significant award for its solution, see [7]. By ageneral result proved in [16], the limit of the ratio ex k − ( n, K k ) n k − as n tends to infinity exists. This is a positive number called the Tur´an density of K k . Let d k = d ( K k ) denote this number, which is conjectured to be 5 / d k is not known, we can prove the following. Theorem 1.1
For any fixed c < d k there is some b > so that h k − ( n, c ) ≤ n b/ log log n . Note that by the results of [11] on supersaturated hyperghraphs if c > d k then any ( k − n vertices with at least cn k − edges contains Ω( n k ) copies of K k . Therefore, for any such c thereis a constant b = b ( c ) > h k − ( n, c ) ≥ bn , implying that the d k bound in Theorem 1.1 istight.Our protocols also imply an extension of the main result of [1]. That is, it entails a constructionof nearly complete ( k − k − k − n vertices cannot contain more than k − (cid:0) nk − (cid:1) < n k − edges, and hence any ( k − bn k − edges cannot be partitioned into less than Θ( n ) partial Steiner systems. Thehypergraph removal lemma shows here, too, that in fact the number of such systems cannot beΘ( n ), that is, for any fixed positive b , this number divided by n must tend to infinity with n . Thefollowing result shows, however, that this number can be smaller than n ǫ for any positive ǫ . Theorem 1.2
For every integer k ≥ , there is an absolute constant c > so that for sufficientlylarge n there is a ( k − -graph on n vertices with at least (1 − o (1)) (cid:18) nk − (cid:19) edges, whose edges can be decomposed into at most n c/ log log n induced subgraphs, each being apartial Steiner system. The rest of the paper contains the proofs of the above two theorems. The organization isas follows. Section 2 contains background on communication complexity and high-dimensional ermutations, 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 n vertices and Ω( n / log n ) edges,decomposable into n O (1 / log log n ) induced matchings. Section 3 contains the application of thisrecipe to prove Theorem 1.1 and Theorem 1.2. The details of the graphs and hypergraphs producedby this recipe, and the proof that it works correctly are given in Section 4. The final Section 5contains a brief summary. General notation
We let [ n ] = { , , . . . , n } . A k -tuple is denoted either ( x , . . . , x k ) or inabbreviated form ~x . 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 cansee [17] for a more comprehensive survey. In the NOF model k players wish to compute a function f : X × X × · · · × X k → { , } . The players agree on a communication protocol P . Then, an input( x , x , . . . , x k ) is presented to the players so player i sees all input except x i , we sometimes referto this player as the x i -player. The players take turns to write messages on a blackboard accordingto the agreed protocol P . Each message of each player may depend on the part of the input seenby this player, and except for the last player it can also depend on the messages written so far onthe blackboard. The message written by the last player depends only on the part of the input hesees, and is independent of the content of the blackboard. One way to visualize this is as if thelast player wrote a message first and then did not participate in the rest of the transaction. Thevalue of the function can be computed by all players from the content of the board at the end ofthe protocol. The cost of a protocol, denoted C ( P ), is the maximal number of bits written on theboard, over all inputs, by the first k − .The string of bits written on the blackboard for a given input ~x = ( x , . . . , x k ) is called a transcript , denoted T ( ~x ). We let T i ( ~x ) for i = 1 , . . . , k be the part of this transcript that is writtenby player i . Let T be a transcript, the subset S = S ( T ) of entries satisfying T ( ~x ) = T and f ( ~x ) = 1,is called a cylinder intersection . Note that a cylinder intersection is defined with respect to afunction and a protocol for this function, we specify the function and protocol when it is necessaryfor a clear presentation and otherwise omit them. In the basic communication complexity definition all players can see each others messages, and the cost of the protocoldepends 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 usual definition of cylinder intersection is more general, what we defined here is referred to as a 1-monochromaticcylinder intersection. Since we are only interested in 1-monochromatic cylinder intersections we abbreviate the notation. e say that a subset of entries S is symmetric if membership in S does not depend on theorder of the first k − S is symmetric if ( x , . . . , x k − , x k ) ∈ S if and only if( x π (1) , . . . , x π ( k − , x k ) ∈ S for every permutation π on { , , . . . , k − } . High-dimensional permutations A line in [ n ] k is a subset L ⊂ [ n ] k such that k − L are fixed, and the remaining coordinate takes all possible values. Following is asimple example with n = 5 and k = 3: L = { (1 , , , (1 , , , (1 , , , (1 , , , (1 , , } . In this example the first and third coordinates are fixed, and the second coordinate takes all possiblevalues in [5] = { , , , , } . There is a distinct line for every choice of unconstrained coordinate i ∈ [ k ], and a choice of values to fix the remaining coordinates. A line in [ n ] × · · · × [ n k ] is definedsimilarly. We say that the line is in the i th dimension if the unconstrained coordinate is i .A ( k − -dimensional permutation is a function f : [ n ] k → { , } such that for every line L in [ n ] k there is exactly one ~x ∈ L such that f ( ~x ) = 1. A sub-permutation is a function f :[ n ] k − × [ N ] → { , } such that every line in the k th dimension contains a single 1, and every otherline contains at most one 1.For example, let G be a group, define f : G k → { , } by f ( x , . . . , x k ) = 1 if and only if x + x + · · · + x k = 0. Then f is a permutation. Let H be a subset of G , then the function h : H k − × G → { , } defined similarly to f , is a sub-permutation.A weak permutation is a function f : [ n ] k → { , } such that every line contains at most one1-entry, and a weak sub-permutation is defined similarly: it is an f : [ n ] k − × [ N ] → { , } with N ≥ n such that every line contains at most one 1-entry. Ruzsa-Szemer´edi graphs and hypergraphs
As mentioned in the introduction, a graph isan ( r, t )-Ruzsa-Szemer´edi graph if its set of edges can be partitioned into t pairwise disjoint inducedmatchings, each of size r . Such a graph obviously has rt edges. A challenge in constructing Ruzsa-Szemer´edi graphs is to make the density of edges as large as possible while keeping the number ofmatchings relatively low. We are therefore less concerned with the size of each matching, and onlyworry about the number of matchings and the density of the edges.There is a natural way to extend the notion of Ruzsa-Szemer´edi graphs to hypergraphs, byconsidering Steiner systems S ( k − , k − Steiner system S ( t − , t ) in a set V , is a familyof t -element subsets of V (called blocks ) such that each ( t − V is containedin exactly one block. A partial Steiner system is defined similarly with the exception that each( t − V is contained in at most one block.For a natural number k >
2, and a ( k − G = ( V, E ) we are interested in partitioning E into induced partial Steiner systems S ( k − , k − V is the set of vertices of a raph, then a partial Steiner system S (1 ,
2) in V is a matching. Thus, this definition extends thenotion of a Ruzsa-Szemer´edi graph. Given a function f : [ n ] k − × [ N ] → { , } , a protocol P for f , and a transcript T of the last player,denote S k ( T ) = { ( x , . . . , x k ) ∈ [ n ] k − × [ N ] : T k ( x , . . . , x k ) = T and f ( x , . . . , x k ) = 1 } . Next we describe a recipe for generating Ruzsa-Szemer´edi graphs and hypergraphs, as well as upperbounds on h k − ( n, c ), from NOF protocols. Recipe 1 - from protocols to graphs and hypergraphs
1. Choose a weak sub-permutation f : [ n ] k − × [ N ] → { , } , for natural numbers n , N and k > .2. Construct a communication protocol P for f .3. Pick a transcript T of the last player so that S k ( T ) is symmetric, and let S = S k ( T ) . The following theorem describes the outcome when following Recipe 1.
Theorem 2.1
Let P be a protocol found in the second step of Recipe 1, and let S be the subset ofinputs picked in the last step. Let p = | S | /n k − , γ = C ( P ) and N ′ = N · γ , then1. There is an (explicitly defined) ( k − -graph on n vertices whose edge density is p , that is theunion of N ′ induced partial Steiner systems S ( k − , k − .2. If p = 1 − o (1) , then h k − ( n, c ) ≤ ( N ′ /n ) for c < d k . Here, the construction of the ( k − -graph that gives the bound is also explicit, given explicit constructions of ( k − -graphs ofdensity d k − o (1) which contain no K k . We defer the proof of Theorem 2.1 and the explicit definition of the graphs produced by Recipe 1to 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.
We apply Theorem 2.1 to prove:
Lemma 2.2
There is a graph on n vertices with edge density Ω(1 / log n ) that is the union of n / Ω(log log n ) induced matchings. roof We follow the steps of Recipe 1:
Choosing the function
Let q, d > n = q d , and define Z q,d = { ( x + y ) : x, y ∈ [ q ] d } . Denote by g q,d : ([ q ] d ) × Z q,d → { , } the function satisfying g q,d ( x, y, z ) = 1if and only if x + y = 2 z (here addition is in R d ). It is not hard to verify that g q,d is a sub-permutation. Denote N = N q,d = | Z q,d | , then N ≤ (2 q ) d = q d · d = n / log q . Since log log n = log d + log log q , we have that N ≤ n / Ω(log log n ) as long as d ≤ q c for someconstant c . We will later choose d = q . The protocol
Next we present a protocol for g q,d . Protocol 1
A protocol for g q,d
1. The z -player computes k x − y k , and writes the result on the board.2. The y -player writes iff k x − y k = 4 k x − z k .3. The x -player writes iff k x − y k = 4 k y − z k . At the end, all players know the value of the function. Indeed, the value of the function is 1 ifthe last two bits written on the board are both equal to 1, and 0 otherwise.
The cost of the protocol
The cost of the protocol is C ( P ) = 2, as the first two players sendonly 2 verification bits. The choice of S By the Chernoff-Hoeffding’s inequality (c.f., e.g., [2]), the quantity k x − y k computed by the third player satisfies P ( (cid:12)(cid:12) k x − y k − E ( k x − y k ) (cid:12)(cid:12) ≥ t ) ≤ e − t dq . Thus, with constant probability, k x − y k takes one of √ dq values. There is, therefore, atranscript T for the third player such that | S ( T ) | ≥ Ω( n / √ dq ). If we take d = q we get | S ( T ) | ≥ Ω( n /d ) ≥ Ω( n / log n ). The fact that S ( T ) is symmetric is easy to verify. Lemma 2.2now follows from Theorem 2.1, part 1.Note that we could improve the density of the graph in Lemma 2.2 to Ω(log log n/ log ǫ n ) forany constant ǫ > / d = q c for an appropriately chosen large constant c . This seemsto be the best one can get when using Protocol 1 though. In the next section we use a variant of his protocol in which the first two players participate more, in order to save communication bitsof the last player. This will allow us to increase the density to near optimal. k = 3 Choosing the function
The function we choose is g q,d , defined in Section 2.3. We later fix d = q . The protocol
For a natural number r let G r = ( V, E r ) be the graph with V = [ q ] d , where d iseven, and E r = { x, y : k x − y k ≤ r } (later we take r = √ d ). The players agree on a proper coloring χ of G r by d r + 1 colors, where d r is its maximum degree. Let µ = E ( k x − y k ) = d ( q − P of [0 , dq ] into intervals of length r + O (1). The playerschoose P that satisfy: the number of intervals in the partition is ⌈ dq /r ⌉ , and the number µ is inthe middle of the interval containing it. As an example, the players can choose a partition whichis a translation of the partition induced by DIV ( L ) = ⌊ Lr ⌋ . Let I r : [0 , dq ] → { , , . . . , dq /r } map a number in [0 , dq ] to the index of the interval containing it, according to P . Given an input( x, y, z ), the players then use the following protocol: Protocol 2
A protocol for g q,d
1. The z -player writes I r ( k x − y k ) on the board.2. The y -player verifies that I r ( k x − y k ) = I r (4 k x − z k ) , and writes on the boardiff this is the case.3. The x -player verifies that I r ( k x − y k ) = I r (4 k y − z k ) , and writes on the boardiff this is the case.4. If one of the last two bits are equal to , reject and finish.5. The x -player writes χ (2 z − y ) on the board.6. The y -player writes the value of g q,d ( x, y, z ) . Theorem 3.1
Protocol 2 is correct.
For the proof of correctness, we use the following two observations (used also in [1]): emma 3.2 (Parallelogram law) Let x, y, z ∈ R d then: k x − y k + k x + y − z k = 2 k x − z k + 2 k y − z k Lemma 3.3 ([1])
For an even integer d > , the number of integral points contained in the ballof radius r in R d is at most: π d/ ( r + 0 . d ( d/ < (2 πe ) d/ ( r + 0 . √ d ) d ( d ) d/ Proof [of Theorem 3.1] By Lemma 3.3, the maximum degree of G r is at most d r = (2 πe ) d/ ( r + 0 . √ d ) d ( d ) d/ . The chromatic number of G r is therefore at most d r + 1.If x + y = 2 z then obviously the protocol reaches step 5. On the other hand, if the protocolreached step 5 then k x − y k , 4 k x − z k , and 4 k y − z k , all lie in the same interval of length r .Thus, by the Parallelogram law k x + y − z k = 2 k x − z k + 2 k y − z k − k x − y k = 12 (cid:0) k x − z k + 4 k y − z k (cid:1) − k x − y k ≤ r . Thus, (2 z − y ) is in a ball B ( x, r ) of radius r around x . Every other vector v ∈ B ( x, r ) is in distanceat most 2 r from (2 z − y ). The color of (2 z − y ) in this ball is therefore unique. It follows that atstep 6 the y -player knows the value of y and hence knows everything. The cost of the protocol
The number of bits used by the first two players is:log d r + Θ(1) = Θ d + d log 2 r + 0 . √ d √ d ! . If we take r = √ d , the cost of the protocol is therefore bounded by C ( P ) ≤ O ( d ) = O (cid:18) log n log q (cid:19) . The choice of S A transcript T of the z -player corresponds to a message I r ( k x − y k ). Thesize of S ( T ) is therefore equal to the number of pairs x, y ∈ [ q ] d satisfying I r ( k x − y k ) = T . oeffding’s inequality implies that P ( (cid:12)(cid:12) k x − y k − µ (cid:12)(cid:12) ≥ t ) ≤ e − t dq . In particular, the probability that I r ( k x − y k ) = I r ( µ ) is at least (1 − e − r dq ) since we chose thepartition of the intervals so that µ lies in the middle of the interval containing it.Take r = √ d , and pick S = S ( T ) for T = I √ d ( µ ), we have | S | ≥ (1 − e − d q ) n . Conclusion
When applying Theorem 2.1 the parameters that we get are: • p = (1 − e − d q ), • N ′ = n / Ω(log log n ) O ( d ) .Taking d = q , and observing that S is symmetric, this proves the k = 3 case of Theorems 1.1 and1.2. k > Choosing the function
Let Z m,q,d = { m ( P mi =1 x i ) : x i ∈ [ q ] d } and define g k,q,d : ([ q ] d ) k − × Z k − ,q,d → { , } by g k,q,d ( x , . . . , x k ) = 1 if and only if x + · · · + x k − = ( k − x k . It is easy toverify that g k,q,d is a sub-permutation, and | Z k − ,q,d | ≤ ( kq ) d = n / log k q . The protocol
The protocol is a simple reduction to the case k = 3. Protocol 3
A protocol for g k,q,d
1. The first player writes on the board if and only if (( k − x k − x − · · · − x k − ) ∈ Z ,q,d .2. If the last bit was equal to , the protocol ends with rejection.3. Players , and k run Protocol 2 for g ,q,d with r = √ d on input x ′ = x , y ′ = x ,and z ′ = (( k − x k − x − · · · − x k − ) . The correctness of the above protocol follows from the correctness of Protocol 2 and the factthat the equation x + x + x + · · · + x k − = ( k − x k holds if and only if x + x = 2( (( k − x k − x − · · · − x k − )). Note that the last equation cannot hold if (( k − x k − x − · · · − x k − )does not belong to Z ,q,d . he cost of the protocol Outside the reduction to Protocol 2, the players send only onemore bit. The cost of the protocol thus satisfy C ( P ) ≤ O ( d ) ≤ O ( log n log q ), as before. The choice of S We can choose, as in Section 3.1, the set S = S k ( T ) for T = I √ d ( µ ). ByHoeffding’s inequality, the size of S is (1 − o (1)) n k − as long as d >> q . The only problem is that S is not symmetric. To remedy that, just add to the protocol a test whether I r ( k x i − x j k ) = I r ( µ )for every 1 ≤ i < j < k . These tests can all be carried out by the last player, so this adds only onemore communication bit, which for simplicity we assume is the last bit. Now pick the transcript T ′ = ( T,
1) which imply that I r ( k x i − x j k ) = I r ( µ ) for all 1 ≤ i < j < k . The corresponding set S k ( T ′ ) is now symmetric, and as long as k is a constant, Hoeffding’s inequality still implies thatthe size of S k ( T ′ ) is at least (1 − o (1)) n k − . We first rephrase Theorem 2.1 slightly.
Theorem 4.1
Let f : [ n ] k − × [ N ] → { , } be a weak sub-permutation, and let S be a symmetriccylinder intersection (w.r.t. f ). Let p = | S | /n k − , then1. There is an (explicitly defined) ( k − -graph on n vertices whose edge density is p , that is theunion of N induced partial Steiner systems S ( k − , k − .2. If p = 1 − o (1) , then h k − ( n, c ) ≤ ( N/n ) for c < d k . Here, the construction of the ( k − -graph that gives the bound is explicit, given explicit constructions of ( k − -graphs of density d k − o (1) which contain no K k . 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 thesubset S . In Theorem 2.1 S is defined by S = S k ( T k ) = { ( x , . . . , x k ) ∈ [ n ] k − × [ N ] : T k ( x , . . . , x k ) = T k and f ( x , . . . , x k ) = 1 } , for some transcript T k of the last player . In Theorem 4.1 on the other hand, S is a cylinderintersection, that is S = S ( T ) = { ( x , . . . , x k ) ∈ [ n ] k − × [ N ] : T ( x , . . . , x k ) = T and f ( x , . . . , x k ) = 1 } , for some transcript T of all players .This difference is easily bridged though. Let f : [ n ] k − × [ N ] → { , } be a weak sub-permutation, P a protocol for f , T k a transcript of the last player, and S = S k ( T k ) a subset, ound using Recipe 1. Let γ = C ( P ), and denote N ′ = N · γ . For simplicity identify [ N ′ ] with[ N ] × { , } γ .Define g : [ n ] k − × [ N ′ ] → { , } by g ( x , . . . , x k − , ( x k , T ...k − )) = 1 if and only if f ( x , . . . , x k − , x k ) =1 and T ...k − = T ( x , . . . , x k ) ◦ · · · ◦ T k − ( x , . . . , x k ). That is, T ...k − is the message written onthe board by the first k − P , on input ( x , . . . , x k ).It is not hard to verify that g is a weak sub-permutation. We use the following protocol P ′ for g , on input ( x , . . . , x k − , ( x k , T ...k − )): the last player sends his message as in P , then each of theother players verifies (using one bit of communication each) that his part in T ...k − agrees with P .Obviously P ′ is correct if and only if P is correct. The subset S ′ = { ( x , . . . , ( x k , T ...k − )) ∈ [ n ] k − × [ N ′ ] : T k ( x , . . . , x k ) = T k and f ( x , . . . , x k ) = 1 } is a cylinder intersection with respect to P ′ and g , and | S ′ | /n k − = | S | /n k − . Theorem 4.1 cannow 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 ofgraphs ( k = 3) and then explain the necessary adjustments for the general case ( k ≥ k = 3 We prove the first conclusion of Theorem 4.1, concerning Ruzsa-Szemer´edi graphs, in Section 4.1.1.The upper bound on h ( n, c ) is proved in Section 4.1.2. We use the following simple fact proved in[18]. Lemma 4.3 ([18])
Let f : [ n ] × [ n ] × [ N ] → { , } be a function satisfying that every line in thethird dimension contains at most a single , and let S be a cylinder intersection (w.r.t f ). Then, S does not contain stars : triplets of the form ( x ′ , y, z ) , ( x, y ′ , z ) , ( x, y, z ′ ) where x = x ′ , y = y ′ and z = z ′ . The relation between Ruzsa-Szemer´edi graphs and the communication complexity of 2-dimensionalpermutations was observed in [18]. The graphs constructed in [18] are bipartite though, and weneed slightly different settings. Let S ⊆ [ n ] × [ n ] × [ N ] be symmetric, define E S = { ( x, y ) , ( x, z ) , ( y, z ) : ( x, y, z ) ∈ S } . Let G S = ( V, E S ) be the graph with vertex set V = V A ∪ V B , where V A = [ n ] and V B = [ N ],and edge set E S . We allow self loops in E S , and consider a collection of self loops as a matching.Note that when S is a cylinder intersection with respect to a weak sub-permutation there is always t most one edge between a pair of vertices. The following lemma implies the first conclusion inTheorem 4.1. Lemma 4.4
Let f : [ n ] × [ n ] × [ N ] → { , } be a weak sub-permutation, and let S be a symmetriccylinder intersection. Let H = ([ n ] , F ) be the subgraph of G S induced on V A . That is: F = { ( x, y ) : ∃ z ∈ V B s.t. ( x, y, z ) ∈ S } . Then, the edges of | F | can be partitioned into N induced matchings. Proof
Partition the edge set F as follows, for every z ∈ B let F z = { ( x, y ) : ( x, y, z ) ∈ S } . This is a partition of F since f a sub-permutation, and therefore there is at most a single z suchthat ( x, y, z ) ∈ S for every ( x, y ) ∈ [ n ] .The fact that F z is an induced matching follows from Lemma 4.3. Assume in contradiction that F z is not an induced matching, then there is an edge ( x, y ) ∈ F z ′ for z ′ = z such that ( x, y ′ ) , ( x ′ , y )are in F z . We then get a star ( x ′ , y, z ) , ( x, y ′ , z ) , ( x, y, z ′ ) ∈ S , contradicting Lemma 4.3. Note thatthe fact that f is a sub-permutation also implies that x ′ = x and y ′ = y . h ( n, c ) Consider the same graph G S as in the previous section. A basic observation is: Lemma 4.5
Let f : [ n ] × [ n ] × [ N ] → { , } be a function satisfying that every line in the thirddimension contains at most a single , and let S be a symmetric cylinder intersection (w.r.t f ).Then, a triangle ( x, y, z ) where x, y ∈ V A and z ∈ V B exists in G S if and only if ( x, y, z ) ∈ S . Proof
The fact that a triangle ( x, y ) , ( x, z ) , ( y, z ) where x, y ∈ V A and z ∈ V B exists in G S forevery ( x, y, z ) ∈ S follows immediately from the definition of E S . Assume in contradiction thatthere is also such a triangle in G S for ( x, y, z ) S . Then necessarily there are x ′ , y ′ ∈ V A and z ′ ∈ V B such that ( x ′ , y, z ) , ( x, y ′ , z ) , ( x, y, z ′ ) ∈ S . But then S contains a star, in contradiction toLemma 4.3. Lemma 4.6
Let f : [ n ] × [ n ] × [ N ] → { , } be a weak sub-permutation, and let S be a symmetriccylinder intersection satisfying | S | = (1 − o (1)) n . Then h ( n, c ) ≤ N /n for c < / . roof Consider the graph G S again. By lemma 4.5, and the fact that f is a weak sub-permutation,an edge in G S appears in exactly one triangle ( x, y, z ) with x, y ∈ V A and z ∈ V B . Therefore, ifwe take a bipartite subgraph inside V A , we will have every edge lie in exactly one triangle, whichis optimal. But, the density of edges in G S is relatively small, since there are n + N vertices andorder of (1 − o (1)) n edges. To remedy this, we define a product function, aiming to increase thedensity of edges. The price we pay is that the number of triangles an edge can lie in increases.Let t ≥ f t : ([2 t ] × [ n ]) × [ N ] → { , } by f (( α, x ) , ( β, y ) , z ) = 1if and only if f ( x, y, z ) = 1. Let S t = { (( α, x ) , ( β, y ) , z ) : ( x, y, z ) ∈ S } . It is not hard to verify that S t is a symmetric cylinder intersection with respect to f t . By Lemma 4.5a triangle (( α, x ) , ( β, y ) , z ) where ( α, x ) , ( β, y ) ∈ ([2 t ] × [ n ]) and z ∈ [ N ] exists in G S t if and onlyif ( x, y, z ) ∈ S . Thus, every edge of G s t lies in at most 2 t triangles of this sort. To remove otherkind of triangles let H = ([2 t ] × [ n ] , E H ) be a bipartite graph with density 1 /
4. Now define E ′ S t = { (( α, x ) , ( β, y )) , (( α, x ) , z ) , (( β, y ) , z ) : ( x, y, z ) ∈ S, (( α, x ) , ( β, y )) ∈ E H } . Then every edge in E ′ S t lies in at least one triangle and at most 2 t triangles. The number of edgessatisfy | E ′ S t | ≥ (1 − o (1))(2 t n ) /
4. The density of edges is thus(1 − o (1)) 14 (2 t n ) (2 t n + N ) . If we take t = 2 log( N/n ) this becomes(1 − o (1)) 14 ( N /n ) ( N /n + N ) . Recall that S is a cylinder intersection of size (1 − o (1)) n . It therefore follows from the graphremoval lemma (and the hypergraph removal lemma for larger k ) - see Theorem 34 in [18] fordetails - that necessarily n = o ( N ). The density is thus (1 − o (1)) . Since every edge is in at most2 t = N /n triangles, this completes the proof. We outline the proof of Theorem 4.1 for k ≥
3. Since the general case is very similar to the proofof the k = 3 case, we do not repeat all the details here.For ~x = ( x , . . . , x k ) ∈ [ n ] k − × [ N ] denote by [ ~x ] k − the family of all subsets of size k − ntries of ~x . That is: [ ~x ] k − = (cid:18) { x , . . . , x k } k − (cid:19) . Let S ⊆ [ n ] k − × [ N ] be a symmetric subset of entries, define E S = [ ~x ∈ S [ ~x ] k − . Let G S = ( V, E S ) be the ( k − V = V A ∪ V B , where V A = [ n ] and V B = [ N ], and edge set E S .The generalized version of Lemma 4.3 is: Lemma 4.7 ([18])
Let f : [ n ] k − × [ N ] → { , } be a function satisfying that every line in the k th dimension contains at most a single , and let S be a cylinder intersection (w.r.t f ). Then, S does not contain stars : k entries of the form ( x ′ , x , . . . , x k ) , ( x , x ′ , . . . , x k ) , ( x , x , . . . , x ′ k ) where x ′ i = x i for i = 1 . . . k . This immediately gives:
Lemma 4.8
Let f : [ n ] k − × [ N ] → { , } be a function satisfying that every line in the k thdimension contains at most a single , and let S be a symmetric cylinder intersection (w.r.t f ).Then, we have that [ ~x ] k − is a copy of K k in G S with x , . . . x k − ∈ V A and x k ∈ V B , if and onlyif ~x = ( x , . . . , x k ) ∈ S . Proof
Similar to the proof of Lemma 4.5, but using Lemma 4.7 instead of Lemma 4.3.The following two lemmas generalize Lemma 4.4 and Lemma 4.6:
Lemma 4.9
For an integer k ≥ , let f : [ n ] k − × [ N ] → { , } be a weak sub-permutation, andlet S be a symmetric cylinder intersection. Let G ′ = ([ n ] , E ′ ) be the subrgraph of G S induced on V A . Then, the edges of | E ′ | can be partitioned into N partial Steiner systems S ( k − , k − . Proof
The proof is similar to the proof of Lemma 4.4, we rewrite the main points. The edges of G ′ are: E ′ = { ( x , . . . , x k − ) : ∃ x k ∈ V B s.t ( x , . . . , x k − , x k ) ∈ S } . Partition the edge set E ′ as follows, for every z ∈ V B let E ′ z = { ( x , . . . , x k − ) : ( x, . . . , x k − , z ) ∈ S } . This is a partition of E ′ since f a sub-permutation, and the fact that E ′ z is a partial Steiner systemfollows from Lemma 4.7. emma 4.10 For an integer k ≥ , let f : [ n ] k − × [ N ] → { , } be a weak sub-permutation, and let S be a symmetric cylinder intersection satisfying | S | = (1 − o (1)) n k − . Then h k − ( n, c ) ≤ ( N/n ) for c < d k . Proof
The proof is very similar to the proof of Lemma 4.6, just instead of taking the subgraph H = ([2 t ] × [ n ] , E H ) to be a bipartite graph with density 1 /
4, take a subhypergraph with no copiesof K k and density d k . Note that we do not need to know d k or H , we just need to know that d k isfinite and that H exists. As mentioned in the introduction, there is a link between the main construction of [1] and theoriginal construction of Ruzsa and Szemer´edi [21]. We describe this link here, starting with a newconstruction, equivalent to the one of Ruzsa and Szemer´edi, derived using the recipe in Section 2.2.Our approach avoids the use of Behrends construction of a large set of integers without a three-termarithmetic progression [3], which was the heart of the construction of Ruzsa and Szemer´edi.
Lemma 5.1 ([21])
There exists a graph on n vertices, with n / O ( √ log n ) edges, that is the unionof Θ( n ) induced matchings. Proof
We follow the steps of Recipe 1. The details are very similar to those in Section 2.3, withslight modifications.
Choosing the function
Let q, d > n = q d . Let f q,d :([ q ] d ) → { , } be the function satisfying f q,d ( x, y, z ) = 1 if and only if x + y = 2 z . It is not hardto verify that f q,d is a weak sub-permutation, in fact it is a weak permutation. We later set q tobe even and d = log( q ) = Θ( √ log n ). The protocol
The protocol is identical to the protocol for g q,d in Section 2.3. The cost of the protocol
The cost of the protocol is C ( P ) = 2. The choice of S By Hoeffding’s inequality, with constant probability, k x − y k takes oneof √ dq values. There is, therefore, a transcript T for the third player such that | S k ( T ) | ≥ Ω( | f − q,d (1) | / √ dq ). Where | f − q,d (1) | is the number of 1’s of the function f q,d . That is, it is thenumber of x, y ∈ [ q ] d such that ( x + y ) / q ] d . Assume for simplicity that q is even, then | f − q,d (1) | ≥ q d · ( q/ d . Therefore | S k ( T ) | ≥ Ω( q d · ( q/ d / √ dq ) ≥ Ω( n / d √ dq ) . aking d = log q = Θ( √ log n ) we get | S k ( T ) | ≥ n / O ( √ log n ) . S k ( T ) is symmetric, thus Lemma 2.2follows from Theorem 2.1.We can now describe the relation between the construction of Ruzsa and Szemer´edi [21] andthat of [1]. Call the construction above A , the simple construction of Section 2.3 B , and theconstruction of Section 3.1 (providing the graphs similar to [1]) C . The table below compares theseconstructions. A B CFunction Domain: ([ q ] d ) Def. rule: x+y=2z Dom.: ([ q ] d ) × Z q,d Def. rule: x+y=2z Dom.: ([ q ] d ) × Z q,d Def. rule: x+y=2zProtocol idea Third player sends k x − y k . Third player sends k x − y k . Third player sendssome bits of k x − y k , then the firsttwo players computethe rest.Number of ver-tices n = q d n = q d n = q d Edge density 2 − O ( √ log n ) Ω(log log n/ log ǫ n )for any constant ǫ > / n ) n O (1 / log log n ) n O (1 / log log n ) References [1] N. Alon, A. Moitra and B. Sudakov, Nearly complete graphs decomposable into large inducedmatchings and their applications, Proc. of the 44 th ACM STOC (2012), 1079-1089. Also: J.Eur. Math. Soc. (JEMS) 15 (2013), no. 5, 1575–1596.[2] N. Alon and J. Spencer, The Probabilistic Method (4th edition), Wiley Interscience, 2016.[3] F. A. Behrend. On sets of integers which contain no three terms in arithmetic progression,Proc. National Academy of Sciences USA 32 (1946), 331–332.[4] Y. Birk, N. Linial, and R. Meshulam. On the uniform-traffic capacity of single-hop inter-connections employing shared directional multichannels.
IEEE Transactions on InformationTheory , 39(1):186–191, 1993.[5] F. R. K. Chung and R. L. Graham,
Erd˝os on Graphs; his legacy of unsolved problems , A KPeters, Ltd., 1998.
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