Point-Hyperplane Incidence Geometry and the Log-Rank Conjecture

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In this work we present several linear-algebraic, incidence-geometric, and set-theoretic conjectures which are connected to the “log-rank conjecture” in communication complexity. We also describe some mild progress on the incidence-geometric questions. We start with some background on communication complexity and incidence geometry.

1.1 Motivation: Communication complexity and the log-rank conjecture The (deterministic) communication complexity of a (two-party) function f : 0, 1 , as defined by Yao [Yao79], measures how much communication is needed for two cooperatingX×Y p →{arties,} one knowing x and the other knowing y , to jointly determine f(x, y). The (deterministic) communication ∈ Y ∈ Y complexity CCdet(f) is the minimum over all communication protocols that compute f(x, y) of the maximum communication over all pairs (x, y) . ∈X×Y Every function f : 0, 1 corresponds naturally to a Boolean matrix Mf , with rows indexed by and columns by X×Y→{, where (M )} = f(x, y). Given a function f : 0, 1 , we can define its X Y f x,y X×Y→{ } rank rank(f) as the rank of Mf over R, which is a linear-algebraic measure of f’s complexity. This leads to a natural question: How is rank(f) connected to CCdet(f)? A monochromatic rectangle for a function f : 0, 1 is a pair , such that f(a,b) is constant over all (a,b) . A c-bit communicationX×Y→{ protocol} partitionsA ⊆ X theB ⊆ space Y into a disjoint union of at most 2c∈monochromatic A×B rectangles. Since monochromatic rectangles correspondX × Y to rank- 1 matrices, log2(rank(f)) CCdet(f)[MS82]. The log-rank conjecture of Lovász and Saks [LS88] posits a converse up to a polynomial≤ factor; that is: Conjecture 1.1 (Log-rank conjecture [LS88]). For every function f : 0, 1 , X×Y→{ } CC (f) polylog(rank(f)). det ≤ This conjecture is a central and notorious open question in communication complexity. Currently, the best known bound for arbitrary f is due to Lovett [Lov16], who proved the following: Theorem 1.2 ([Lov16]). For every function f : 0, 1 , X×Y→{ } CC (f) O rank(f) log(rank(f)) . det ≤ p The log-rank conjecture asserts that every low rank Boolean matrix can be partition into a small number of monochromatic rectangles. An obviously necessary condition for this is the presence of a large monochromatic rectangle. A result due to Nisan and Wigderson [NW95] shows that this is in fact also a sufficient condition. Specifically, define the size of a rectangle ( , ) as . Then: A B |A||B| Theorem 1.3 ([NW95], as articulated in [Lov16]). Suppose that there exists some function γ : N N such that the following is true: For every Boolean function f : 0, 1 , f contains a monochromatic→ γ(rank(f)) X×Y →{ } rectangle of size at least 2− . Then for every Boolean function f : 0, 1 , |X ||Y| · X×Y→{ } log(rank(f)) r CC (f) O log2(rank(f)) + γ . det ≤  2i  i=0 X   In particular, proving that the hypothesis of this theorem holds with γ(d) = polylog(d) would suffice to prove the log-rank conjecture (Conjecture 1.1).1

1This reduction is tight in a strong sense: A c-bit protocol for f partitions X ×Y into ≤ 2c monochromatic rectangles, one c of which must have size at least |X||Y|· 2− .

2 1.2 Background: Incidence geometry In this section, we give various definitions and notations that we will use throughout the rest of the paper. In Rd, a hyperplane is the locus of points x Rd defined by an equation of the form a, x = b, for some a = 0 Rd,b R. We refer to the vector a as∈ the normal vector of h and b as its offseth (denotedi b(h)); a 6 ∈ ∈ hyperplane is homogeneous if it has offset zero. A pair of hyperplanes h,h′ are parallel if for some constant c R we have a = ca′ (where h and h′ are defined by a, x = b and a′, x = b′, respectively). ∈ A point p and a hyperplane h in Rd are incident ifhp liesi on h; weh calli the pair (p,h) an incidence (and say p is incident to h and vice versa). We refer to a collection of points together with a collection of hyperplanes (in the same ambient space Rd) as a configurationP. Configurations determine an incidenceH graph G( , ), an (unweighted, undirected) bipartite graph defined as follows: The left vertices are the points ,P theH right vertices are the hyperplanes , and the edge (p,h) is included iff p is incident to h. FollowingP Apfelbaum and Sharir [AS07], we denoteH by I( , ) the total number of incidences between and (equiv., the number of edges in G( , )) and by rs(P ,H) the largest number of edges in any completeP Hbipartite subgraph of G( , ); the readerP H may verify theP equivalentH characterization that P H rs( , )= max ( p : p lies on S h : S lies on h ).2 P H S affine subspace Rd |{ ∈P }| · |{ ∈ H }| ⊂ I( , ) rs( , ) Let = n and = m; we refer to the ratios PmnH and mnP H as the incidence and bipartite subgraph densities|P| of the configuration|H| ( , ), respectively. The rank of a matrix has aP naturalH incidence-geometric interpretation: Given a matrix of rank d, every row corresponds to a point in Rd and every column to a hyperplane in Rd, and the entry of the matrix determines how much one needs to shift the hyperplane so that it contains the point. The hope that there is a large monochromatic rectangle in a matrix translates in this interpretation to the hope that there is a large bipartite subgraph in the point-hyperplane incidence graph, or equivalently that there is an affine subspace that is contained in many hyperplanes and contains many points. Moreover, the fact that the matrices under consideration are Boolean matrices implies that these point-hyperplane configurations have an unusually large density of incidences (specifically 50% of the point-hyperplane pairs are incident!). One could ask if simply the high density of incidences suffices to imply the existence of a large bipartite subgraph polylog(d) (of density 2− in d dimensions). This is known to be false and a construction of Lovett [Lov16] Θ(√d) 3 with density 2− is a counterexample; see §5 for discussion on this and related constructions. In Conjecture 2.8 (which is an incidence-geometric analogue of [Lov16, Conjecture 5.1]), we hypothesize that these counterexamples are the “worst possible”, i.e., in a configuration with incidence density Ω(1), the e O(√d) bipartite subgraph density is at least 2− .

1.3 Contributions Connecting the log-rank conjecture to large bipartite subgraphs in “structured” configurations. In view of the above, we consider what additional properties of the point-hyperplane configurations could potentially lead to the presence of a large bipartite subgraph. Building on unpublished work of Golovnev, Meka, Sudan and Velusamy [GMSV19], we note that Boolean matrices lead to configurations where the hyperplanes can be partitioned into pairs of parallel hyperplanes such that each pair covers the set of points. This leads to a easy reformulation of the log-rank conjecture (Conjecture 2.4). We extend this formulation to the notion of parallel k-partitions (see Definition 2.1) that allow the sets of hyperplanes to be partitioned into sets of size k for an arbitrary constant k. In Theorem 2.10, we show that the log-rank conjecture is equivalent to the assertion (Conjecture 2.6) that the k-parallel partitionable configurations contain large bipartite subgraphs. Theorem 2.10 has an natural interpretation as a reduction from bounds for parallel k-partitioned configurations to (stronger) bounds for (k 1)-partitioned configurations. In the matrix corresponding to a point-hyperplane configuration,− there is a natural linear-algebraic prop- erty equivalent to the presence of a parallel k-partition, which we refer to as k-listability: A matrix is k-listable if every column has at most k distinct entries (though these sets may differ arbitrarily across

2The original motivation for the notation rs(P, H): r refers to the quantity |{p ∈ P : p lies on S}| and s refers to the quantity |{h ∈ H : S lies on h}|; rs(P, H) maximizes the product rs over all aﬃne subspaces of Rd. 3Though admittedly the authors were not aware of this at earlier stages of this writing [SS21].

3 columns). Theorem 2.10 can hence be stated linear-algebraically, in closer spirit to the original log-rank conjecture: the log-rank conjecture is equivalent to the assertion that every k-listable matrix contains a large 1-listable submatrix. (See §2 for a more careful account of the connection between matrices, configurations, listability, and parallel partitionability.) We believe that the covering aspect of the point-hyperplane incidences is a key element of the log- rank conjecture and posit an extension (which does not immediately seem to be equivalent to the log-rank conjecture, nor does it seem to have a simple linear-algebraic formulation). Specifically in Conjecture 2.7 we suggest that if a set of hyperplanes can be partitioned (in a not-necessarily-parallel way) into blocks of size at most k such that each covers a given set of points, then the incidence graph corresponding to this configuration has a large bipartite subgraph.

New bounds for bipartite subgraph size in general configurations. Returning to the more basic question of the incidence density of a configuration versus the size of its largest bipartite subgraph, we present two results that improve the state of the art. In comparison to previous works (i.e., [AS07]), we do not take the dimension d to be not a constant. Our first result here (Theorem 3.1) is a lower bound on the size of complete bipartite subgraphs in incidence graphs of constant density. Specifically in §3, we give a self-contained probabilistic argument that for a set of n points and a collection of m hyperplanes in Rd, for incidence density ǫ = I( , )/mn, we mustP have rs( , ) Ω(mnH ǫ2d/d). For context, Apfelbaum and Sharir [AS07] proved, usingP H substantial P H ≥ · d 1 incidence-geometric machinery, the bound rs( , ) Ωd(mn ǫ − ) (with an unknown implicit constant depending on dimension). By contrast, our proofP H is≥ elementary,· and in our parameter regime of interest (where incidence density is some constant ǫ and dimension is non-constant), we can conclude the bipartite subgraph density is at least ǫO(d) (which is not implied by the result of [AS07], due to the implicit constant). Theorem 3.1 also recovers the incidence-geometric fact that, fixing d and ǫ, we have rs( , ) Ωǫ,d(mn); as observed by Apfelbaum and Sharir [AS07, p. 3], this behavior is not present in randomP bipartiteH ≥ graphs with edge density, say, ǫ =1/2. O(d) On the flip side, in Theorem 4.1 we give an explicit construction which has rs( , ) O(mn 2− ) and I( , ) = Θ(mn 1/√d). Since the incidence density of this construction isP notH constant,≤ ·it is too small toP falsifyH the most· general conjecture (Conjecture 2.8), which posits that if the incidence density is constant, there is a somewhat large bipartite subgraph. Theorem 4.1 is proven by modifying by interpreting the lattice-based construction of Apfelbaum and Sharir [AS07] probabilistically and applying concentration bounds, in order to gain a factor of √d in I( , ). (That is, Apfelbaum and Sharir [AS07]’s construction has only I( , )=Θ(mn/d). Other explicit constructions,P H discussed in §5, have I( , ) Ω(mn) but only P H O(√d) P H ≥ rs( , ) O(mn 2− )[Lov16; Lov21; Pál21; FW21].) P H ≤ · Explicit constructions of configurations without large bipartite subgraphs, and products of Boolean matrices. In §5, we analyze an explicit point-hyperplane configuration which was suggested as a counterexample by Pálvölgyi [Pál21]; our analysis demonstrates that O(√d) is the best possible exponent in Conjecture 2.8. (A different counterexample with the same quantitative parameters and a related analysis, due to Lovett [Lov16] and also indicated by Fox and Wigderson [FW21], was already known.) More generally, we observe that these constructions all arise from products of Boolean matrices, and we pose special cases of our general conjectures for these types of matrices as interesting variants. Since they involve products of Boolean matrices, we also describe them in the language of set systems and extremal combinatorics.

Discussion. We believe that our new incidence-geometric and linear-algebraic frameworks for interpreting the log-rank conjecture shed light on several important prior results: in particular, the positive results of [Lov16] (i.e., Theorem 1.2), which in our incidence-geometric language is a bipartite subgraph density lower e O(√d) bound of 2− for parallel 2-partitioned configurations, and the constructions of [Lov16; Lov21; Pál21; Ω(√d) FW21] of (unstructured) configurations with incidence density Ω(1) and bipartite subgraph density 2− . For context, Lovett’s analysis [Lov16] relies heavily on the binarity of the matrix; roughly, monochromatic rectangles are created by using a hyperplane rounding argument that exploits the gap between the two possible values for entries in the matrix. Altogether we are left in the following unsettling situation: The O(√d) only way we know to prove a 2− bound uses binarity, but we believe that we should be able to get (1)

4 polylog(d) O(√d) a 2− bound using binarity (i.e., Conjecture 2.4) and (2) a 2− bound without using structural assumptions (i.e., Conjecture 2.8). This situation is especially interesting in light of the reduction used to prove Theorem 2.10. According f(d) to this reduction, if we have a 2− bipartite subgraph density lower bound for parallel (k 1)-partitioned 2 (f(d)) − configurations, then we also have a 2− bound for parallel k-configurations. In particular, since the e O(√d) best bound we know for parallel 2-partitioned configurations is 2− (i.e., Theorem 1.2 due to [Lov16]), e O(d) the best bound we know for parallel 3-partitioned configurations is 2− — which simply recovers what we already proved in Theorem 3.1! On the other hand, a modest improvement in the bounds for parallel 2-partitioned configurations would yield a nontrivial bound for parallel 3-partitioned configurations. While this is potentially simply due to a technical weakness of the reduction, it may still help explain the difficulty in suprassing the “√d barrier”.

1.4 Notation We follow notation which is mostly derived from Apfelbaum and Sharir’s work [AS07]. denotes a set of points and a set of hyperplanes, and n = and m = .4 In real space Rd, a j-flat Pis a j-dimensional affine subspace.H Hence, a point is a 0-flat and|P| a hyperplane|H| is a (d 1)-flat. A configuration ( , ) of points and hyperplanes determines a point-hyperplane incidence graph −G( , ), which is a bipartiteP H graph with left-vertex set , right-vertex set , and with an edge (p,h) iff p is incidentP H to h. I( , ) denotes the total number of incidencesP between andH (which is equal to the number of edges in G(P ,H )). rs( , ) is the largest number of edges in anyP completeH bipartite subgraph of G( , ). P H P H We employ standard asymptotic notation; we use subscripts toP denH ote arbitrary dependence on implicit constants, e.g., f O 1 (g) implies that there exists a constant C, depending arbitrarily on z ,...,z , ≤ z ,...,zk 1 k such that f(x) Cg(x) for sufficiently large x. We use f O(g) to denote “f O(g polylog(g))”. We use [n] to denote the≤ set of integers 1,...,n . ≤ ≤ · { } A submatrix of or rectangle in a matrix M RX ×Y ise given by two subsets and and denoted M . We use these terms interchangeably.∈ A ⊆ X B ⊆ Y |A×B 2 Incidence-geometric reformulations of the log-rank conjecture

In this section we present some reformulations of the log-rank conjecture in terms of incidence geometric questions. The conjectures start with some unpublished work of Golovnev, Meka, Sudan and Velusamy [GMSV19] who raised Conjecture 2.4 below explicitly and also went on to propose a stronger form of Conjecture 2.8 (which was [SS21, Conjecture 5]). The latter turns out to be false (and this was already known — see [Lov16] and §5), and so we propose several new variants here and prove some equivalences.

2.1 The “original” reformulation To begin, we introduce new notions of structured point-hyperplane configurations. Definition 2.1 (Parallel k-partition). Let ( , ) be a point-hyperplane configuration. A parallel k-partition for ( , ) is a partition of into disjoint blocksP H of size k such that (1) within each block , P H H H1 ⊔···⊔Hℓ Hi the hyperplanes are all mutually parallel, and (2) for each block i and point p , p is incident to one of the hyperplanes of . H ∈P Hi Note that in a parallel k-partitioned configuration, every point is incident to precisely one hyperplane in each block. Next, we define properties of matrices which we will soon show are analogous to parallel k-partitionability: n m Definition 2.2 (k-listability and k-arity). A matrix M R × is k-listable if every column of M contains at most k distinct entries. Moreover, it is k-ary if M contains∈ at most k total distinct entries. Note that 1-listability is equivalent to every column being constant, and that k-arity implies k-listability.

4Our use of n to denote the number of points and m to denote the number of hyperplanes is opposite to Apfelbaum and Sharir [AS07]. Also, Apfelbaum and Sharir [AS07] use Π instead of H to denote the hyperplane-set.

5 Next, we describe a natural correspondence between parallel k-partitioned configurations and k-listable matrices. Specifically, we define a matrix associated with every configuration, and conversely, a configuration associated with every matrix. d Given a configuration ( , ) of n points and m hyperplanes in R with a parallel k-partition 1 ℓ, let Mat( , ) be the n mP matrixH defined as follows: Let p be the i-th point in and let aHbe⊔···⊔H the unit P H × i P j normal vector corresponding to the hyperplanes in block j . Then the (i, j)-th entry of Mat( , ) is pi,aj . H n m P H h i Now we describe the configuration Con(M) associated with a matrix M R × of rank d. Let M = PQ n d d m d ∈ d where P R × and Q R × . Let p ,...,p R denote the rows of P and let q ,...,q R denote ∈ ∈ 1 n ∈ 1 m ∈ the columns of Q. For j [m] let Bj denote the set of distinct entries in column j of M. For each j [m] b ∈ d ∈ and b Bj , define hj as the hyperplane determined by the equation x, qj = b over x R . We define the configuration∈ h i ∈ Con(M) := ( p : i [n] , hb : j [m],b B ). { i ∈ } { j ∈ ∈ j } The basic facts about the correspondence between matrices and configurations are summarized in Proposition 2.3 below. Given a set of hyperplanes, define the offset set ( ) R as ( ) := b(h): h . H B H ⊆ B H { ∈ H} Proposition 2.3. 1. If ( , ) is a parallel k-partitioned configuration, then Mat( , ) is k-listable and ( ) -ary and has rankP Hd. P H |B H | ≤ n m 2. If a matrix M R × is k-listable and k′-ary and has rank d, then Con(M) has a parallel k-partition, ∈ (Con(M)) k′, and Con(M) contains mk hyperplanes. |B |≤ 3. If ( , ) is a parallel k-partitioned configuration, then Mat( , ) has a 1-listable submatrix of size at leastP rs(H , ) (and hence a monochromatic rectangle of sizeP atH least rs( , )/ ( ) ). P H P H |B H | 4. If M has a monochromatic rectangle of size r, then rs(Con(M)) r. ≥ The proofs follows immediately from the definitions and so we omit them. According to this correspondence and Proposition 2.3, the following conjecture is equivalent to Conjecture 1.1: Conjecture 2.4 (Parallel 2-partitioned configurations have large bipartite subgraphs [GMSV19]). In Rd, let ( , ) be a parallel 2-partitioned configuration with ( )= 0, 1 . Then P H B H { } polylog(d) rs( , ) mn 2− . P H ≥ · Theorem 2.5. The log-rank conjecture (Conjecture 1.1) holds if and only if Conjecture 2.4 does. Proof. (= ) Given any parallel 2-partitioned configuration ( , ) in Rd, we may assemble the matrix Mat( , ⇒). By Proposition 2.3, rank(Mat( , )) d and MatP (H, ) is binary, i.e., Mat( , ) has two P H P H ≤ P H P H distinct entries, a and b. Letting M := (Mat( , ) a)/(b a), we have rank(M) = rank(Mat( , )) (since neither subtracting a constant nor rescaling changesP H − rank). Ass− uming the log-rank conjecture, byP theH converse polylog(d) to Theorem 1.3, M has a monochromaticf rectangle of size 2− f; hence so does Mat( , ), polylog(d) ≥ |P||H| · nP mH so by Proposition 2.3, rs( , ) 2− . ( =) Given any binary matrix M 0, 1 × of P H ≥ |P||H| · ⇐ ∈ { } rank at most d, wef may form the configuration Con( , ) in Rd, which has by Proposition 2.3 a parallel P H polylog(d) 2-partition and 2m hyperplanes. Assuming Conjecture 2.4, rs(Con( , )) 2mn 2− . Hence by P H ≥ ·polylog(d) Proposition 2.3 again, M contains a monochromatic rectangle of size at least mn 2− , which suffices by Theorem 1.3 to prove the log-rank conjecture. ·

2.2 Relaxations of Conjecture 2.4 We could hope to relax the structure we require in Conjecture 2.4, to only require the presence of a parallel partition of fixed size: Conjecture 2.6 (Parallel partitioned configurations have large bipartite subgraphs). The following is true for every fixed integer k> 1. In Rd, let ( , ) be a configuration with a parallel k-partition. Then P H polylog(d) rs( , ) mn 2− . P H ≥ · n m Equivalently, by Proposition 2.3, all k-listable matrices M R × contain 1-listable submatrices of size at polylog(rank(M)) ∈ least mn 2− . ·

6 We will show in §2.3 that Conjecture 2.6 is actually equivalent to Conjecture 2.4 (and thus to the log-rank conjecture). Next, we could relax the parallel requirement of partition: Conjecture 2.7 (Partitioned configurations have large bipartite subgraphs). The following is true for every fixed integer k > 1. In Rd, let ( , ) be a configuration with a (not-necessarily-parallel) k-partition, i.e., P H such that can be partitioned into blocks 1 ℓ of size k such that for each block i and point p , p isH incident to at least one of the hyperplanesH ⊔···⊔H of . Then H ∈P Hi polylog(d) rs( , ) mn 2− . P H ≥ · We are currently unable to show that Conjecture 2.7 is implied by Conjecture 2.6. Ultimately, we might hope to drop the partitioning requirement entirely, and conjecture that for every collection of n points and of m hyperplanes, as long as I( , ) ǫmn for some fixed constant ǫ > 0, P polylog(d) H P H ≥ rs( , ) mn 2− . Indeed the previous version of this paper [SS21] contained such a conjecture, butP thisH is≥ unfortunately· known to be false [Lov16; Lov21; Pál21; FW21]; see §5. The strongest possible version of such a statement that may yet turn out to be true is the following conjecture (which is roughly an incidence-geometric restatement of [Lov16, Conjecture 5.1]): Conjecture 2.8 (Configurations with incidence density Ω(1) have somewhat large bipartite subgraphs). The following is true for every fixed ǫ> 0. In Rd, let be a collection of n points and a collection of m hyperplanes, such that I( , ) ǫ mn. Then P H P H ≥ · e O(√d) rs( , ) mn 2− . P H ≥ · We note that Conjecture 2.8 is too weak to prove the log-rank conjecture (Conjecture 1.1). It would, however, yield a result roughly matching the current best upper bound (Theorem 1.2, due to [Lov16]) on communication complexity as a function of the rank (up to some logarithmic factors).

2.3 Equivalence of Conjecture 2.4 and Conjecture 2.6 In this section, we show that Conjecture 2.6 is implied by Conjecture 2.4.

n m Lemma 2.9 (Folklore). If M R × has rank d, and p R[X] is a real polynomial, then the matrix N given by N = p(M ) for every∈ (i, j) [n] [m] has rank∈ at most dc, where S(p) := c 0 : ij ij ∈ × c S(p) { ≥ p contains a nonzero monomial of degree c . ∈ } P Proof. Recall that rank is subadditive: If A and B are matrices, then rank(A + B) rank(A) + rank(B). c ≤ c Hence it suﬃces to show that for every c, the matrix N given by Nij = Mij has rank at most d . n d d m If M has rank d, we can write M = PQ for some P R × ,Q R × ; let pi and qj denote the i-th row ∈ ∈ c of P and the j-th column of Q, respectively. We have Mij = pi, qj by deﬁnition. Then let pi′ := pi⊗ , i.e., c h i the c-fold self-tensor product of pi, which is the d -dimensional vector whose entries correspond to products c of each possible sequence of c elements of pi. Similarly, let qj′ := qj⊗ . Hence we have

d c c c N = M = p , q = p q = p 1 q 1 p c q c = p′ , q′ , ij ij h i j i i,k j,k i,k j,k ··· i,k j,k h i j i k=1 ! k1,...,kc [d] X X∈ c where p and q denote the k-th entries of p and q , respectively. Hence letting P Rn d be the matrix i,k j,k c i j ′ × d m ∈ whose i-th row is pi′ and Q′ R × be the matrix whose j-th column is qj′ , we have N = P ′Q′, so N has rank at most dc. ∈ Theorem 2.10. If the log-rank conjecture holds (in the form of Conjecture 2.4), then Conjecture 2.6 holds. In particular, assuming Conjecture 2.4, for all integers k > 2, there exists a polynomial p such that the k ′ ′ n m n m following is true: Every k-listable, rank-d matrix M R × has a 1-listable submatrix M ′ R × of size pk(log d) ∈ ∈ m′n′ mn 2− . ≥ ·

7 Proof. Firstly, we argue that it is enough to prove the theorem only for matrices M which (1) have no 1-listable (i.e., constant) columns and (2) contain a 0 and 1 in every column. Firstly, we reduce to the case where (1) holds: If at least half of M’s columns are 1-listable, then we immediately have a 1-listable submatrix of M containing all the rows and at least half the columns. Otherwise, we may throw out all the 1-listable columns, thereby reducing the total number of columns by at most half. Next, we reduce to the case where (2) holds as well. Since (1) holds, we can let aj ,bj j [m] with aj = bj be such that j-th n m { } ∈ 6 column of M contains aj and bj . Let A R × be the rank-1 matrix with column j being the constant m m ∈ vector (aj ,...,aj ). Let D R × be the diagonal matrix with (j, j)-th entry being 1/(bj aj ). Now let N := (M A)D. Then rank(∈ N) rank(M A) d + 1 by subadditivity of rank, and moreover− every column of −N contains some 0 entry≤ and some− 1 entry.≤ And proving the theorem for N immediately implies the theorem for M, since if S [n],T [m] are such that N S T is a 1-listable submatrix of N, then ⊆ ⊆ | × (N + AD) S T is 1-listable and hence so is M S T . Now, we| × will prove the theorem by induction| × on k. The k = 2 case is implied by the log-rank conjecture (Conjecture 1.1), since if every column of M is 2-listable and contains a 0 and 1, M is precisely a Boolean matrix. (Let p2 be the polynomial given by the log-rank conjecture.) x x For general k, assume the theorem holds for k 1, and let pk(x) satisfy pk(x) pk 1(log((2 + 1)(2 + − 2 ≥ − 2))) + p2(x) for all x 1 (e.g., pk(x) = pk 1((x + 1)) + p2(x) will suﬃce). For an arbitrary k-listable, ≥ − rank-d matrix M with a 0 and a 1 in every column, let M be the matrix with M := M (M 1). M ij ij ij − has rank at most (d + 1)(d +2) by Lemma 2.9. Also M is (k 1)-listable (since 0 and 1s of M became f − f f 0 in M). So by induction there is a submatrix of M given by rows S and columns T such that M S T is × pk−1(log(d+1)(d+2)) | 1-listable, and S T mn 2 . Now MfS T is 2-listable, since each column of M S T is | || | ≥ · | × | × somef constant value c, and any value in the correspondingf column of M S T must be a root of z(zf 1) = c. | × − Moreover, rank(M S T ) rank(M)= d, since the rank of a submatrix never exceeds the originalf matrix’s | × ≤ rank. We thus conclude, now using the base case k = 2, that there exist S′ S and T ′ T with M S′ T ′ p2(log d) ⊆ ⊆ p2(log| ×d) being 1-listable and S′ T ′ S T 2− . Combining the above we have S′ T ′ mn 2− pk−1(log(d+1)(d+2)) | || pk|(log ≥d |) || | · | || | ≥ · · 2− 2− . ≥ 3 Exponential lower bound on bipartite subgraph density using probabilistic method

We use the probabilistic method to prove the following: Theorem 3.1. Let and be a set of n points and m hyperplanes, respectively, in Rd, such that I( , ) d P H ǫ P H ≥ ǫmn. If n is sufficiently large (in particular, if 2 n> 1), then ǫ2d rs( , ) Ω mn . P H ≥ d We rely on the following standard fact, which the reader may verify: Proposition 3.2. In Rd, let S be a j-flat and h a hyperplane. Suppose that S h = and h S. Then S h is a (j 1)-flat. ∩ 6 ∅ 6⊇ ∩ − That is, the operation of “nontrivial intersection with a hyperplane” reduces the dimension of a flat by one. Proof of Theorem 3.1. Consider the following randomized process for choosing an affine subspace S: Select H1,...,Hd uniformly and independently from , and output S := H1 Hd. Let manypoints denote d ǫ H ∩···∩ the event “at least 2 -fraction of the points in are incident to S”. For j [d], let badplanej denote d P ∈ ǫ the event “at least 3d -fraction of the hyperplanes in contain H1 Hj 1 and Hj does not contain H ∩···∩ − d H1 Hj 1”. (In the case j = 1, we define the empty intersection as all of R , so that badplane1 never occurs.)∩···∩ − Observe that for any fixed point p , and for each j [d], p is incident to Hj with probability ǫ. Hence by independence, p is incident∈ P to S with probability∈ ǫd. So by a Markov-type argument, ≥ ≥

8 d ǫ Pr[manypoints] 2 . Moreover, for each j [d], since Hj is independent of H1,...,Hj 1, Pr[badplanej ] d ≥ ∈ − ≤ ǫ 3d . Hence, by the union bound, the probability of the event “manypoints doesn’t occur or badplanej occurs d for some j” is at most 1 ǫ < 1. So there exists a list of hyperplanes h ,...,h such that manypoints occurs − 6 1 d and none of the events badplanej occur. But it cannot be the case that for all j, hj h1 hj 1, since then Proposition 3.2 implies that S is either a point or empty, so manypoints cannot6⊇ occur∩···∩ by assumption.− d d ǫ ǫ Hence for some j, Prh [h h1 hj 1] 3d , and correspondingly, Prh [h S] 3d . Hence S is d ∼H d − ∼H ǫ ⊇ ∩···∩ǫ ≥ ⊇ ≥ incident to at least 2 n points and 3d m hyperplanes, as desired.

4 Explicit upper bound construction with exponentially small bipartite subgraphs but sub-constant incidence density

In this section, we present a lattice-based explicit upper bound construction, which modifies an upper bound construction of Apfelbaum and Sharir [AS07, Theorem 1.3] (itself based on ideas from Elekes and Tóth [ET05]). More precisely, we construct configurations ( , ) of points and hyperplanes with expo- Pd H nentially small bipartite subgraphs (i.e., rs( , )/mn O(2− )). Such configurations would almost be a counterexample to Conjecture 2.8, except thatP H the fraction≤ of incidences I( , )/mn = Θ(1/√d) vanishes as d , whereas a counterexample to Conjecture 2.8 would need I( , )/mnP H Ω(1). Our→ ∞ result is as follows: P H ≥ Theorem 4.1. For every d> 0, there exists a set of n points and a set of m hyperplanes in Rd such Pd H that I( , ) Ω(nm/√d) and rs( , ) O(mn 2− /√d). P H ≥ P H ≤ · The only difference between Theorem 4.1 and Apfelbaum and Sharir’s result [AS07, Theorem 1.3] is the gain of a √d factor in the lower bound on I( , ), which will be achieved by exhibiting a dense subset of Apfelbaum and Sharir’s construction (see ClaimP H 4.5 below). Specifically, we will construct configurations with n = Θ(2d√d) points, m = Θ(2d) hyperplanes, I( , )= Θ(22d) incidences, and a bipartite subgraph upper bound rs( , ) O(2d). P H P H ≤ Proof of Theorem 4.1. Assume for simplicity that d 1 is a perfect square which is divisible by 4. Consider the set of points − d 1 d 1 := (x1,...,xd): x1,...,xd 1 0, 1 , xd − √d 1,..., − + √d 1 P − ∈{ } ∈ 4 − − 4 − and the set of hyperplanes

d := x R : aixi =0 : a1,...,ad 1 0, 1 ,ad = 1 . H ∈ − ∈{ } − (( i ) ) X d 1 d d 1 d By construction, n = = 2 − 2√d +1 = Θ(2 √d) and m = = 2 − = Θ(2 ). Also deﬁne the |P| · |H| “universe” of points

:= (x1,...,xd): x1,...,xd 1 0, 1 , xd 0,...,d 1 . U { − ∈{ } ∈{ − }} d 1 contains and numbers 2 − d points in total. U ApfelbaumP and Sharir [AS07,· pp. 16-17] proved the following three claims: 2d 2 Claim 4.2. I( , )=2 − . U H d d j 1 Claim 4.3. Let f R be a j-ﬂat. Then f intersects in at most 2 − − points. ⊂ U Claim 4.4. Let f Rd be a j-ﬂat that is contained in some hyperplane h . f is contained in at most 2j hyperplanes of ⊂. ∈ H H The latter two claims together imply that rs( , )= O(2d); we include proofs of all three in Appendix A for completeness. Finally, we have: U H

9 2d 2 d Claim 4.5. I( , ) (1 2/e)2 − = Θ(2 ). P H ≥ − Proof. We proceed probabilistically, showing that “many” settings of the values x1,...,xd 1,a1,...,ad 1 d 1 − − result in a value for x = − a x that lies within the interval [(d 1)/4 √d 1, (d 1)/4+ √d 1]. d i=1 i i − − − − − Consider the following experiment: Choose x1,...,xd 1,a1,...,ad 1 uniformly and independently from P d 1 − − 0, 1 , and output Succeed if the sum i=1− aixi lies in the aforementioned interval. { } d 1 Note that since each a x Bern(1/4) independently, the expectation of − a x is (d 1)/4. Then i i P i=1 i i the Chernoﬀ bound gives us ∼ − P d 1 − Pr a x (d 1)/4 √d, (d 1)/4+ √d 2exp (1/√d)2d =2/e. i i 6∈ − − − ≤ − "i=1 # X h i Thus, the experiment outputs Succeed with probability at least 1 2/e, which is constant. Hence 2d 2 − I( , ) (1 2/e) I( , )=(1 2/e)2 − . P H ≥ − U H − Claim 4.5 suﬃces to prove the theorem, since we have also that rs( , ) rs( , ). U H ≥ P H 5 Discussion: Upper bounds from cross-intersecting families, and why Conjecture 2.8 is necessarily weak

In this section, we report on constructions [Lov16; Lov21; Pál21; FW21] which show that O(√d) is the best possible exponent we could hope for in Conjecture 2.8, i.e., we exhibit explicit configurations with e O(√d) rs( , )/mn 2− and I( , )/mn Ω(1) (see Theorem 5.3 below). These constructions all arise fromP productsH ≤ of Boolean matricesP ,H and there≥ are a number of natural questions in this area which we pose. A length-d Boolean vector can be viewed as the indicator of a subset of [d], and using the language of set systems will provide another helpful perspective on the log-rank conjecture. (Using this perspective to construct counterexamples was suggested by [Lov21] and [FW21].) Let , ([d]) be two set systems on [d]. Following are two notions which describe patterns among the intersectionA B⊆P sizes A B for A ,B . For ǫ [0, 1], we say that , are ǫ-almost cross-disjoint | ∩ | ∈ A ∈ B ∈ A B if PrA ,B [A B = ] ǫ, and exactly cross-disjoint in the special case ǫ = 0. Following [Sne03], for L 0∼A,...,d∼B , we∩ say6 that∅ ≤ , are L-cross-intersecting if for every A and B , A B L.5 ⊆{These notions} have linear-algebraicA B interpretations. For , ∈ A([d]), we∈B can| define∩ | the∈ matrix A B ⊆ P Mat( , ) 0,...,d A×B whose (A, B)-th entry is A B . , are ǫ-almost cross-disjoint iff all but ǫ-fractionA B of∈Mat { ( , )’s} entries are zeros. , are L-cross-intersecting| ∩ | A B iff Mat( , ) is L -ary. We have two conjecturesA B about pairs of setA systems.B The first conjecture wouldA beB implied| | by Conjecture 2.8: Conjecture 5.1. The following is true for every fixed ǫ > 0. Let , ([d]) be ǫ-almost cross-disjoint. e A B⊆P O(√d) Then there exists , such that and are exactly cross-disjoint, and 2− . e R ⊆ A S⊆B R S |R||S|O(√d) ≥ |A||B| · Equivalently, Mat( , ) contains a 0-monochromatic rectangle of density at least 2− . A B The second conjecture would be implied by Conjecture 2.6: Conjecture 5.2. The following is true for every fixed ℓ > 0. Let , ([d]) be L-cross-intersecting, where L = ℓ. Then there exists , , and t L, such thatA forB⊆P all A ,B , A B = t, and | | polylog(d) R ⊆ A S⊆B ∈ ∈ R ∈ S | ∩ | 2− . Equivalently, Mat( , ) contains a monochromatic rectangle of density at least |R||S|polylog( ≥ |A||B|d) · A B 2− . Lovett [Lov21] independently suggested studying the special case of Conjecture 5.2 where ℓ = 2, which is perhaps the simplest combinatorial version of the log-rank conjecture. (He notes that in the subcase where ℓ =2 and 0 L, Conjecture 5.2 is known to hold, since the “log-nonnegative-rank conjecture” is known to hold (and rescaling∈ to get a Boolean matrix).) The following example based on the idea of Pálvölgyi [Pál21] shows the necessity of the exponent O(√d) in Conjecture 2.8 and Conjecture 5.1.

5However, we do allow 0 ∈ L, in contrast to e.g. [Sne03].

10 Theorem 5.3. The following is true for every ǫ in a dense subset of (0, 1). There exists an infinite, increasing sequence of dimensions d , d ,..., and an infinite sequence of set systems ( , ) on [d ], such 1 2 Ai Bi i that ( i, i) is δi-almost cross-disjoint with δi ǫ as i , but Mat( i, i) contains no 0-monochromatic A B Ω(√d) → → ∞ A B submatrices of density larger than 2− . 1/α Proof. Consider any positive rational number α; we will prove the theorem for ǫ := e− (note that the 1/α image of Q (0, 1) under the map α e− is dense in (0, 1) by continuity). Consider,∩ in increasing order, all7→ values b N such that a := αb is also an integer (there are infinitely many such b by rationality). Let d := ab. We∈ will identify subsets of [d ] with [a] [b], i.e., with a b i i × × Boolean matrices. Let i ([di]) be the subset of matrices which have exactly one 1 in every column, and A ⊂P b let i = i. If n := i and m := i , then we have n = m = a . b B A |A | |B | b (a 1) Each A i is disjoint from (a 1) sets in i; as b (and hence di) approaches , the ratio −m ∈ A 1/α − B ∞ approaches the constant e− . Moreover, consider any 0-monochromatic rectangle i, i. Defining R∗ := R R and R ⊂ A S ⊂B ∈R S∗ := S S, we see that R∗ and S∗ must be disjoint. Hence we may assume without loss of generality ∈S S that is the set of all matrices supported on R∗ and the set of all matrices supported on S∗, and that RS S R∗ and S∗ are complementary. Defining sj := ([a] j ) (i.e., the size of R∗’s support in column j), |R ∩ ×{ } | a we see that = s1 sb and = (a s1) (a sb). This product is maximized when each sj = 2 ; |R| a 2b··· |S| − ··· − hence 2 . (This analysis also leads to a monochromatic rectangle with this size, assuming a is even.) |R| · |S| ≤ An alternative proof was given by Lovett [Lov16; Lov21] and Fox and Wigderson [FW21]. This proof still sets = but instead picks the sets in randomly with some appropriate sparsity, and deduces a e BO(√dA) A similar 2− bound. Finally, we remark that although intersecting and cross-intersecting set systems are widely studied in extremal combinatorics, typically the goal in the literature is to bound the largest possible sizes of set systems with particular combinatorial properties. However, results in extremal combinatorics still have some interesting consequences towards our conjectures. For instance, Snevily [Sne03] showed that in the L case = ([d]) which are L-cross-intersecting and 0 L, we must have = O(d| |). In linear- algebraicA terms,B⊆P if M = XXT is ℓ-ary where X is Boolean6∈ and of size n d,|A| we must|B| have ≤ d Ω(n1/ℓ). × ≥ Acknowledgements

We would like to thank Sasha Golovnev, Raghu Meka and Santhoshini Velusamy for permission to describe their work [GMSV19] here. We would also like to thank Shachar Lovett for his comments [Lov21] on the earlier version of this paper [SS21] and for his counterexample to our original Conjecture 5, as well as Dömötor Pálvölgyi, and Jacob Fox and Yuval Wigderson for their counterexamples to the same [Pál21; FW21]. We thank Dömötor Pálvölgyi for his generous permission to build on his example in §5. Finally, we acknowledge helpful comments from anonymous reviewers which improved the exposition of this paper.

References

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12 A Proofs of claims from Apfelbaum and Sharir [AS07]

In this appendix, for completeness, we include proofs due to Apfelbaum and Sharir [AS07] of claims which together bound rs( , ) (in the context of the proof of Theorem 4.1). Claim 4.2 has a short proof: U H Proof of Claim 4.2. Consider any fixed hyperplane h . Since ad = 1, for any values x1,...,xd 1 ∈ H − d 1 − ∈ 0, 1 , there is a unique value xd 0,...,d 1 such that i aixi = 0, i.e., xd = i=1− aixi. Thus, { } d 1 d 1 ∈ { 2d−2 } h =2 − . Since =2 − , =2 − . |U ∩ | |H| |U ∩ H| P P We need a bit more setup to prove Claim 4.3 and Claim 4.4. For fixed h , consider the linear map ∈ H d 1 d 1 − φ : R − h : (x1,...,xd 1) x1,...,xd 1, aixi . → − 7→ − i=1 ! X φ is an isomorphism whose inverse is a projection onto all but the last coordinate. Restricting φ to the d 1 lattice 0, 1 − , we get a bijection with the points h. We also have the following helpful proposition: { } U ∩ ′ ′ ′ Proposition A.1. Let f Rd be a j-flat. Then I(f, [k]d ) kj . In particular, I(f, 0, 1 d ) 2j . ⊂ ≤ { } ≤ ′ Proof sketch. Verify inductively on d by splitting the lattice [k]d into a disjoint union of parallel sublattices: ′ ′ ′ ′ d d 1 d 1 [k] = ( 1 [k] − ) ( k [k] − ). { }× ⊔···⊔ { }× These ideas let us prove the two remaining claims.

1 d 1 Proof of Claim 4.3. The map φ is an isomorphism, so the preimage φ− (f) is a j-flat in R − . By Proposition A.1, 1 d 1 j d 1 I(φ− (f), 0, 1 − 2 . Since φ restricts to a bijection between 0, 1 − and h, and f , I(f, ) 2{j . } | ≤ { } U ∩ ⊆ U U ≤ Proof of Claim 4.4. Consider any plane h such that h f. Let h be defined by the equation a, x +b =0 d ∈ H ⊇ j d h i d j over x R . View f as the image of an affine injection ψ : R R given by x Mx + v, where M R × has rank∈ j, and v Rd. → 7→ ∈ Since h f, for∈ any y Rj , a,My + v + b = 0. Hence a, v + b = 0 (plugging in y = 0), so a,My =0 ⊇j ∈ h i h i j h j i for all y R (subtracting). Now a⊤M is simply a vector in R (or more precisely, a vector in R ’s dual ∈ j space); if its inner product with all y R is zero, then it is zero. Thus, a ker(M ⊤). ∈ ∈ Let K := ker(M ⊤). By rank-nullity, and since row-rank equals column-rank,

dim(K)= d dim(im(M ⊤)) = d dim(im(M)) = d j. − − −

Now consider the hyperplane h′ defined by the equation x = 1. By definition a h′, so a K h′. But d − ∈ ∈ ∩ K h′, since 0 K but 0 h′. Hence K h is a (d j 1)-flat by Proposition 3.2, so by Proposition A.1, 6⊂ ∈ d j 1 6∈ ∩ − − there are at most 2 − − unique values of a. Finally, we observe that given a, we have b = a, v , so a uniquely determines b. −h i