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Zarankiewicz’s problem for semilinear hypergraphs

Published online by Cambridge University Press:  31 August 2021

Abdul Basit
Affiliation:
Department of Mathematics, Iowa State University, Ames, IA50011, USA; E-mail: abasit@iastate.edu.
Artem Chernikov*
Affiliation:
Department of Mathematics, University of California Los Angeles, Los Angeles, CA90095-1555, USA.
Sergei Starchenko
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN46656, USA; E-mail: sstarche@nd.edu.
Terence Tao
Affiliation:
Department of Mathematics, University of California Los Angeles, Los Angeles, CA90095-1555, USA; E-mail: tao@math.ucla.edu.
Chieu-Minh Tran
Affiliation:
Department of Mathematics, University of Notre Dame, Notre Dame, IN46656, USA; E-mail: mtran6@nd.edu.

Abstract

A bipartite graph $H = \left (V_1, V_2; E \right )$ with $\lvert V_1\rvert + \lvert V_2\rvert = n$ is semilinear if $V_i \subseteq \mathbb {R}^{d_i}$ for some $d_i$ and the edge relation E consists of the pairs of points $(x_1, x_2) \in V_1 \times V_2$ satisfying a fixed Boolean combination of s linear equalities and inequalities in $d_1 + d_2$ variables for some s. We show that for a fixed k, the number of edges in a $K_{k,k}$ -free semilinear H is almost linear in n, namely $\lvert E\rvert = O_{s,k,\varepsilon }\left (n^{1+\varepsilon }\right )$ for any $\varepsilon> 0$ ; and more generally, $\lvert E\rvert = O_{s,k,r,\varepsilon }\left (n^{r-1 + \varepsilon }\right )$ for a $K_{k, \dotsc ,k}$ -free semilinear r-partite r-uniform hypergraph.

As an application, we obtain the following incidence bound: given $n_1$ points and $n_2$ open boxes with axis-parallel sides in $\mathbb {R}^d$ such that their incidence graph is $K_{k,k}$ -free, there can be at most $O_{k,\varepsilon }\left (n^{1+\varepsilon }\right )$ incidences. The same bound holds if instead of boxes, one takes polytopes cut out by the translates of an arbitrary fixed finite set of half-spaces.

We also obtain matching upper and (superlinear) lower bounds in the case of dyadic boxes on the plane, and point out some connections to the model-theoretic trichotomy in o-minimal structures (showing that the failure of an almost-linear bound for some definable graph allows one to recover the field operations from that graph in a definable manner).

Type
Discrete Mathematics
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

1. Introduction

We fix $r \in \mathbb {N}_{\geq 2}$ and let $H = \left (V_1, \dotsc , V_r; E \right )$ be an r-partite and r-uniform hypergraph (or just an r-hypergraph for brevity) with vertex sets $V_1, \dotsc , V_r$ having $\lvert V_i\rvert = n_i$ , (hyper-) edge set E and a total number $n = \sum _{i=1}^r n_i$ of vertices.

Zarankiewicz’s problem asks for the maximum number of edges in such a hypergraph H (as a function of $n_1, \dotsc , n_r$ ) assuming that it does not contain the complete r-hypergraph $K_{k, \dotsc , k}$ with $k> 0$ a fixed number of vertices in each part. The following classical upper bound is due to Kővári, Sós and Turán [Reference Kővári, Sós and Turán14] for $r=2$ and Erdős [Reference Erdős9] for a general r: if H is $K_{k, \dotsc , k}$ -free, then $\lvert E\rvert = O_{r,k} \left (n^{r - \frac {1}{k^{r-1}}} \right )$ . A probabilistic construction in [Reference Erdős9] also shows that the exponent cannot be substantially improved.

However, stronger bounds are known for restricted families of hypergraphs arising in geometric settings. For example, if H is the incidence graph of a set of $n_1$ points and $n_2$ lines in $\mathbb {R}^2$ , then H is $K_{2,2}$ -free, and the Kővári–Sós–Turán Theorem implies $\lvert E\rvert = O\left ( n^{3/2}\right )$ . The Szemerédi–Trotter Theorem [Reference Szemerédi and Trotter20] improves this and gives the optimal bound $\lvert E\rvert = O\left (n^{4/3}\right )$ . More generally, [Reference Fox, Pach, Sheffer, Suk and Zahl12] gives improved bounds for semialgebraic graphs of bounded description complexity. This is generalised to semialgebraic hypergraphs in [Reference Do8]. In a different direction, the results in [Reference Fox, Pach, Sheffer, Suk and Zahl12] are generalised to graphs definable in o-minimal structures in [Reference Basu and Raz2] and, more generally, in distal structures in [Reference Chernikov, Galvin and Starchenko4].

A related highly nontrivial problem is to understand when the bounds offered by the results in the preceding paragraph are sharp. When H is the incidence graph of $n_1$ points and $n_2$ circles of unit radius in $\mathbb {R}^2$ , the best known upper bound is $\lvert E\rvert =O\left (n^{4/3}\right )$ , proven in [Reference Spencer, Szemerédi and Trotter19] and also implied by the general bound for semialgebraic graphs. Any improvement to this bound will be a step toward resolving the long-standing unit-distance conjecture of Erdős (an almost-linear bound of the form $\lvert E\rvert =O\left (n^{1+c/\log \log n}\right )$ will positively resolve it).

This paper was originally motivated by the following incidence problem: Let H be the incidence graph of a set of $n_1$ points and a set of $n_2$ solid rectangles with axis-parallel sides (which we refer to as boxes) in $\mathbb {R}^2$ . Assuming that H is $K_{2,2}$ -free – that is, no two points belong to two rectangles simultaneously – what is the maximum number of incidences $\lvert E\rvert $ ? In the following theorem, we obtain an almost-linear bound (which is much stronger than the bound implied by the aforementioned general result for semialgebraic graphs) and demonstrate that it is close to optimal:

Theorem (A).

  1. 1. For any set P of $n_1$ points in $\mathbb {R}^2$ and any set R of $n_2$ boxes in $\mathbb {R}^2$ , if the incidence graph on $P \times R$ is $K_{k,k}$ -free, then it contains at most $O_k \left (n \log ^{4}(n) \right )$ incidences (Corollary 2.38 with $d=2$ ).

  2. 2. If all boxes in R are dyadic (i.e., direct products of intervals of the form $\left [s2^t, (s+1)2^t\right )$ for some integers $s,t$ ), then the number of incidences is at most $O_k \left ( n \frac {\log \left (100+n_1\right )}{\log \log \left (100+n_1\right )} \right )$ (Theorem 4.7).

  3. 3. For an arbitrarily large n, there exists a set of n points and n dyadic boxes in $\mathbb {R}^2$ so that the incidence graph is $K_{2,2}$ -free and the number of incidences is $\Omega \left (n \frac {\log (n)}{\log \log (n)} \right )$ (Proposition 3.5).

Problem 1.1. While the bound for dyadic boxes is tight, we leave it as an open problem to close the gap between the upper and lower bounds for arbitrary boxes.

Remark 1.2. A related result in [Reference Fox and Pach11] demonstrates that every $K_{k,k}$ -free intersection graph of n convex sets on the plane satisfies $\lvert E\rvert = O_{k}(n)$ . Note that in Theorem (B) we consider a $K_{k,k}$ -free bipartite graph, so in particular there is no restriction on the intersection graph of the boxes in R.

Theorem (A.1) admits the following generalisation to higher dimensions and more general polytopes:

Theorem (B).

  1. 1. For any set P of $n_1$ points and any set B of $n_2$ boxes in $\mathbb {R}^d$ , if the incidence graph on $P \times B$ is $K_{k,k}$ -free, then it contains at most $O_{d,k} \left ( n \log ^{2 d} n \right )$ incidences (Corollary 2.38).

  2. 2. More generally, given finitely many half-spaces $H_1, \dotsc , H_s$ in $\mathbb {R}^d$ , let $\mathcal {F}$ be the family of all possible polytopes in $\mathbb {R}^d$ cut out by arbitrary translates of $H_1, \dotsc , H_s$ . Then for any set P of $n_1$ points in $\mathbb {R}^d$ and any set F of $n_2$ polytopes in $\mathcal {F}$ , if the incidence graph on $P \times F$ is $K_{k,k}$ -free, then it contains at most $O_{k,s}\left ( n \log ^{s} n \right )$ incidences (Corollary 2.37).

Problem 1.3. What is the optimal bound on the power of $\log n$ in Theorem (B)? In particular, does it actually have to grow with the dimension d?

Remark 1.4. A bound similar to Theorem (B.1) and an improved bound for Theorem (A.1) in the $K_{2,2}$ -free case are established independently by Tomon and Zakharov in [Reference Tomon and Zakharov22], in which they also use our Theorem (A.3) to provide a counterexample to a conjecture of Alon et al. [Reference Alon, Basavaraju, Chandran, Mathew and Rajendraprasad1] about the number of edges in a graph of bounded separation dimension, as well as to a conjecture of Kostochka from [Reference Kostochka13]. Some further Ramsey properties of semilinear graphs are demonstrated by Tomon in [Reference Tomon21].

The upper bounds in Theorems (A.1) and (B) are obtained as immediate applications of a general upper bound for Zarankiewicz’s problem for semilinear hypergraphs of bounded description complexity.

Definition 1.5. Let V be an ordered vector space over an ordered division ring R (e.g., $\mathbb {R}$ viewed as a vector space over itself). A set $X \subseteq V^d$ is semilinear, of description complexity $(s,t)$ , if X is a union of at most t sets of the form

$$ \begin{align*} \left\{ \bar{x}\in V^{d}:f_{1}\left(\bar{x}\right) \leq 0, \dotsc, f_{p} \left(\bar{x}\right) \leq 0,f_{p+1}\left(\bar{x}\right)<0,\dotsc,f_{s}\left(\bar{x}\right)<0\right\}, \end{align*} $$

where $p \leq s \in \mathbb {N}$ and each $f_i: V^d \to V$ is a linear function – that is, of the form

$$ \begin{align*}f\left(x_{1},\dotsc,x_{d}\right)=\lambda_{1}x_{1}+\dotsb+\lambda_{d}x_{d}+a\end{align*} $$

for some $\lambda _{i}\in R$ and $a\in V$ .

We focus on the case $V=R = \mathbb {R}$ in the introduction, when these are precisely the semialgebraic sets that can be defined using only linear polynomials.

Remark 1.6. By a standard quantifier elimination result [Reference Van den Dries23, §7], every set definable in an ordered vector space over an ordered division ring, in the sense of model theory, is semilinear (equivalently, a projection of a semilinear set is a finite union of semilinear sets).

Definition 1.7. We say that an r-hypergraph H is semilinear, of description complexity $(s,t)$ , if there exist some $d_i \in \mathbb {N}, V_i \subseteq \mathbb {R}^{d_i}$ and a semilinear set $X \subseteq \mathbb {R}^d = \prod _{i \in [r]} \mathbb {R}^{d_i}$ of description complexity $(s,t)$ so that H is isomorphic to the r-hypergraph $\left (V_1, \dotsc , V_r; X \cap \prod _{i \in [r]} V_i \right )$ .

We stress that there is no restriction on the dimensions $d_i$ in this definition. We obtain the following general upper bound:

Theorem (C). If H is a semilinear r-hypergraph of description complexity $(s,t)$ and H is $K_{k, \dotsc , k}$ -free, then

$$ \begin{align*}\lvert E\rvert = O_{r,s,t,k} \left( n^{r-1} \log^{s\left(2^{r-1}-1\right)}(n) \right).\end{align*} $$

In particular, $\lvert E\rvert = O_{r,s,t,k,\varepsilon } \left ( n^{r-1 + \varepsilon } \right )$ for any $\varepsilon>0$ in this case. For a more precise statement, see Corollary 2.36 (in particular, the dependence of the constant in $O_{r,s,t,k}$ on k is at most linear).

Remark 1.8. It is demonstrated in [Reference Mustafa and Pach17] that a similar bound holds in the situation when H is the intersection hypergraph of $(d-1)$ -dimensional simplices in $\mathbb {R}^d$ .

One can get rid of the logarithmic factor entirely by restricting to the family of all finite r-hypergraphs induced by a given $K_{k, \dotsc , k}$ -free semilinear relation (as opposed to all $K_{k, \dotsc , k}$ -free r-hypergraphs induced by a given arbitrary semilinear relation, as in Theorem (C)).

Theorem (D). Assume that $X \subseteq \mathbb {R}^d = \prod _{i \in [r]} \mathbb {R}^{d_i}$ is semilinear and X does not contain the direct product of r infinite sets (e.g., if X is $K_{k, \dotsc , k}$ -free for some k). Then for any r-hypergraph H of the form $\left (V_1, \dotsc , V_r; X \cap \prod _{i \in [r]} V_i \right )$ for some finite $V_i \subseteq \mathbb {R}^{d_i}$ , we have $\lvert E\rvert = O_X\left (n^{r-1}\right )$ .

This is Corollary 5.12 and follows from a more general Theorem 5.6 connecting linear Zarankiewicz bounds to a model-theoretic notion of linearity of a first-order structure (in the sense that the matroid given by the algebraic closure operator behaves like the linear span in a vector space, as opposed to the algebraic closure in an algebraically closed field – see Definition 5.3).

In particular, for every $K_{k,k}$ -free semilinear relation $X \subseteq \mathbb {R}^{d_1} \times \mathbb {R}^{d_2}$ (equivalently, X definable with parameters in the first-order structure $(\mathbb {R}, <, +)$ by Remark 1.6) we have $\lvert X \cap (V_1 \times V_2)\rvert = O(n)$ for all $V_i \subseteq \mathbb {R}^{d_i}_i$ , $\lvert V_i\rvert = n_i$ , $n = n_1 + n_2$ . One the other hand, by optimality of the Szemerédi–Trotter bound, for the semialgebraic $K_{2,2}$ -free point-line incidence graph $X = \left \{(x_1,x_2; y_1, y_2) \in \mathbb {R}^4 : x_2 = y_1 x_1 + y_2 \right \}\subseteq \mathbb {R}^2 \times \mathbb {R}^2$ we have $\lvert X \cap (V_1 \times V_2)\rvert = \Omega \left (n^{\frac {4}{3}}\right )$ . Note that in order to define it we use both addition and multiplication – that is, the field structure. This is not coincidental; as a consequence of the trichotomy theorem in o-minimal structures [Reference Peterzil and Starchenko18], we observe that the failure of a linear Zarankiewicz bound always allows us to recover the field in a definable way (Corollary 5.11). In the semialgebraic case, we have the following corollary that is easy to state (Corollary 5.14):

Theorem (E). Assume that $X \subseteq \mathbb {R}^d = \prod _{i \in [r]} \mathbb {R}^{d_i}$ for some $r,d_i \in \mathbb {N}$ is semialgebraic and $K_{k, \dotsc , k}$ -free, but $\lvert X \cap \prod _{i \in [r]} V_i\rvert \neq O\left (n^{r-1}\right )$ . Then the graph of multiplication $\times \restriction _{[0,1]}$ restricted to the unit box is definable in $(\mathbb {R}, <, +, X)$ .

We conclude with a brief overview of the paper.

In Section 2 we introduce a more general class of hypergraphs definable in terms of coordinate-wise monotone functions (Definition 2.1) and prove an upper Zarankiewicz bound for it (Theorem 2.17). Theorems (A.1), (B) and (C) are then deduced from it in Section 2.5.

In Section 3 we prove Theorem (A.3) by establishing a lower bound on the number of incidences between points and dyadic boxes on the plane, demonstrating that the logarithmic factor is unavoidable (Proposition 3.5).

In Section 4, we establish Theorem (A.2) by obtaining a stronger bound on the number of incidences with dyadic boxes on the plane (Theorem 4.7). We use a different argument, relying on a certain partial order specific to the dyadic case, to reduce from $\log ^4(n)$ given by the general theorem to $\log (n)$ . Up to a constant factor, this implies the same bound for incidences with general boxes when one counts only incidences that are bounded away from the border (Remark 4.8).

Finally, in Section 5 we prove a general Zarankiewicz bound for definable relations in weakly locally modular geometric first-order structures (Theorem 5.6), deduce Theorem (D) from it (Corollary 5.12) and observe how to recover a real closed field from the failure of Theorem (D) in the o-minimal case (Corollary 5.11).

2. Upper bounds

2.1. Coordinate-wise monotone functions and basic sets

For an integer $r\in \mathbb N_{>0}$ , by an r-grid (or a grid, if r is clear from the context) we mean a cartesian product $B=B_1{{\times }\dotsb {\times }} B_r$ of some sets $B_1, \dotsc , B_r$ . As usual, $[r]$ denotes the set $\left \{1, 2, \dotsc , r \right \}$ .

If $B=B_1{{\times }\dotsb {\times }} B_r$ is a grid, then by a subgrid we mean a subset $C \subseteq B$ of the form $C=C_1{{\times }\dotsb {\times }} C_r$ for some $C_i \subseteq B_i$ .

Let B be an r-grid, S an arbitrary set and $f: B \to S$ a function. For $i \in [r]$ , set

$$ \begin{align*}B^{i} = B_1 {{\times}\dotsb{\times}} B_{i-1} \times B_{i+1}{{\times}\dotsb{\times}} B_r,\end{align*} $$

and let $\pi _i: B \to B_i$ and $\pi ^i: B \to B^i$ be the projection maps.

For $a \in B^i $ and $b \in B_i$ , we write $a \oplus _i b$ for the element $c \in B$ with $\pi ^i(c) = a$ and $\pi _i(c) = b$ . In particular, when $i = r$ , $a \oplus _r b = (a, b)$ .

Definition 2.1. Let B be an r-grid and $(S,<)$ a linearly ordered set. A function $f\colon B\to S$ is coordinate-wise monotone if for any $i\in [r]$ , $a,a'\in B^i$ and $b,b'\in B_i$ , we have

$$ \begin{align*} f(a \oplus_i b)\leq f(a \oplus_i b') \Longleftrightarrow f(a'\oplus_i b)\leq f({a'}\oplus_i b'). \end{align*} $$

Remark 2.2. Let $B =B_1{{\times }\dotsb {\times }} B_r$ be an r-grid and $\Gamma $ an ordered abelian group. We say that a function $f\colon B\to \Gamma $ is quasi-linear if there exist some functions $f_i\colon B_i\to \Gamma $ , $i\in [r]$ , such that

$$ \begin{align*} f(x_1,\dotsc,x_r)=f_1(x_1)+\dotsb +f_r(x_r). \end{align*} $$

Then every quasi-linear function is coordinate-wise monotone (as $ f(a\oplus _i b) \leq f({a} \oplus _i b') \Leftrightarrow f_i(b) \leq f_i(b')$ for any $a \in B^i$ ).

Example 2.3. Suppose that V is an ordered vector space over an ordered division ring R, $d_i \in \mathbb {N}$ for $i \in [r]$ , and $f: V^{d_1} {{\times }\dotsb {\times }} V^{d_r} \to V $ is a linear function. Then f is obviously quasi-linear, and hence coordinate-wise monotone.

Remark 2.4. Let B be a grid and $C \subseteq B$ a subgrid. If $f\colon B\to S$ is a coordinate-wise monotone function, then the restriction $f{\restriction C}$ is a coordinate-wise monotone function on C.

Definition 2.5. Let B be an r-grid. A subset $X\subseteq B$ is a basic set if there exists a linearly ordered set $(S,<)$ , a coordinate-wise monotone function $f\colon B\to S$ and $l\in S$ such that $X= \left \{ b\in B \colon f(b) < l\right \}$ .

Remark 2.6. If $r=1$ , then every subset of $B=B_1$ is basic.

Remark 2.7. If $X\subseteq B$ is given by $X= \left \{ b\in B \colon f(b) \leq l\right \}$ for some coordinate-wise monotone function $f\colon B\to S$ , then X is a basic set as well. Indeed, we can just add a new element $l'$ to S so that it is a successor of l, and then $X=\left \{b \in B: f(b)< l'\right \}$ .

Similarly, the sets $\left \{ b\in B \colon f(b)> l\right \}, \left \{ b\in B \colon f(b) \geq l\right \}$ are basic, by inverting the order on S.

We have the following ‘coordinate-splitting’ presentation for basic sets:

Proposition 2.8. Let $B=B_1{{\times }\dotsb {\times }} B_r$ be an r-grid and $X\subseteq B$ a basic set. Then there is a linearly ordered set $(S,<)$ , a coordinate-wise monotone function $f^r\colon B^r \to S$ and a function $f_r\colon B_r\to S$ such that $X=\left \{ b^r \oplus _{r} b_r \colon f^r(b^r) < f_r(b_r) \right \}$ .

Remark 2.9. The converse of this proposition is also true: an arbitrary linear order $(S,<)$ can be realised as a subset of some ordered abelian group $(G, +, <)$ with the induced ordering (we can take $G := \mathbb {Q}$ when S is at most countable); then define $f: B \to S$ by setting

$$ \begin{align*} f(b^r \oplus_r b_r) := f^r(b^r) - f_r(b_r), \text{ and } l := 0. \end{align*} $$

Proof of Proposition 2.8. Assume that we are given a coordinate-wise monotone function $f\colon B\to S$ and $l\in S$ with $X= \left \{ b\in B \colon f(b) < l\right \}$ .

For $i\in [r]$ , let $\leq _i$ be the preorder on $B_i$ induced by f – namely, for $b,b'\in B_i$ we set $b\leq _i b'$ if and only if for some (equivalently, any) $a\in B^i$ we have $f(a \oplus _i b)\leq f(a \oplus _i b')$ .

Quotienting $B_i$ by the equivalence relation corresponding to the preorder $\leq _i$ if needed, we may assume that each $\leq _i$ is actually a linear order.

Let $<^r$ be the partial order on $B^r$ with $(b_1,\dotsc ,b_{r-1}) <^r \left (b^{\prime }_1,\dotsc ,b^{\prime }_{r-1}\right )$ if and only if

$$ \begin{align*} (b_1,\dotsc,b_{r-1}) \neq \left(b^{\prime}_1,\dotsc,b^{\prime}_{r-1}\right) \text{ and } b_j\leq_j b^{\prime}_j \text{ for all } j\in[r-1]. \end{align*} $$

Define $T := B^r \dot \cup B_r$ , where $\dot \cup $ denotes the disjoint union. Clearly $<^r$ is a strict partial order on T – that is, a transitive and antisymmetric (hence irreflexive) relation.

For any $b^r\in B^r$ and $b_r\in B_r$ , we define

$$ \begin{align*} b^r\triangleleft b_r \text{ if } f(b^r \oplus_r b_r) <l \text{, and } b_r\triangleleft b^r \text{ otherwise}. \end{align*} $$

Claim 2.10. Set $a_1,a_2\in B^r$ and $b_1,b_2\in B_r$ .

  1. 1. If $a_1\triangleleft b_1 \triangleleft a_2 \triangleleft b_2$ , then $b_2 <_r b_1$ and $a_1\triangleleft b_2$ .

  2. 2. If $b_1\triangleleft a_1 \triangleleft b_2 \triangleleft a_2$ , then $b_2 <_r b_1$ and $b_1\triangleleft a_2$ .

Proof. (1). We have $f(a_2 \oplus _r b_1) \geq l$ and $f(a_2 \oplus _{r} b_2) < l$ , hence $b_2 <_r b_1$ . Since $f(a_1 \oplus _r b_1)<l$ and $b_2 <_r b_1$ we also have $f(a_1 \oplus _r b_2)<l$ .

(2) is similar.

Let $\triangleleft ^t$ be the transitive closure of $\triangleleft $ . It follows from the preceding claim that $\triangleleft ^t=\triangleleft \cup \triangleleft {\circ }\triangleleft $ . More explicitly, for $b_1,b_2 \in B_r$ , we have $b_1 \triangleleft ^t b_2$ if $b_2 <_r b_1$ , and for $a_1,a_2 \in B^r$ , we have $a_1 \triangleleft ^t a_2$ if $f(a_1 \oplus b) < l < f(a_2 \oplus b)$ for some $b \in B_r$ . It is not hard to see then that $\triangleleft ^t$ is antisymmetric, and hence it is a strict partial order on T.

Claim 2.11. The union $<^r \cup \triangleleft ^t$ is a strict partial order on T.

Proof. We first show transitivity. Note that $<^r$ and $\triangleleft ^t$ are both transitive, so it suffices to show for $x, y, z \in T$ that if either $x <^r y \triangleleft ^t z$ or $x \triangleleft ^t y <^r z$ , then $x \triangleleft ^t z$ . Furthermore, since $\triangleleft ^t=\triangleleft \cup \triangleleft {\circ }\triangleleft $ , we may restrict our attention to the following cases: If $a_1 <^r a_2\triangleleft b$ with $a_1,a_2\in B^r$ and $b\in B_r$ , then $f(a_1 \oplus _r b)<f(a_2 \oplus _r b)<l$ , and so $a_1\triangleleft b$ . If $b\triangleleft a_1 <^r a_2$ with $a_1,a_2\in B^r$ and $b\in B_r$ , then $f(a_2 \oplus _r b)>f(a_1 \oplus _r b)\geq l$ , and so $b\triangleleft a_2$ .

To check antisymmetry, assume $a_1 <^r a_2$ and $a_2 \triangleleft ^t a_1$ . Since $a_1,a_2\in B^r$ , we have $a_2\triangleleft b \triangleleft a_1$ for some $b\in B_r$ . We have $f(a_1 \oplus _r b)\geq l> f(a_2 \oplus _r b)$ , contradicting $a_1<^r a_2$ .

Finally, let $\prec $ be an arbitrary linear order on $T=B^r\dot \cup B_r$ extending $<^r \cup \triangleleft ^t$ . Since $\prec $ extends $\triangleleft $ , for $a\in B^r$ and $b\in B_r$ we have $(a,b)\in X$ if and only if $a\prec b$ .

We take $f^r\colon B^r\to T$ and $f_r\colon B_r\to T$ to be the identity maps. Since $\prec $ extends $<^r$ , the map $f^r$ is coordinate-wise monotone.

2.2. Main theorem

Definition 2.12. Let $B=B_1{{\times }\dotsb {\times }} B_r$ be an r-grid.

  1. 1. Given $s \in \mathbb {N}$ , we say that a set $X\subseteq B$ has grid-complexity s (in B) if X is the intersection of B with at most s basic subsets of B.

    We say that X has finite grid-complexity if it has grid-complexity s for some $s \in \mathbb {N}$ .

  2. 2. For integers $k_1,\dotsc , k_r$ we say that $X\subseteq B$ is $K_{k_1,\dotsc ,k_r}$ -free if X does not contain a subgrid $C_1{{\times }\dotsb {\times }} C_r\subseteq S$ with $\lvert C_i\rvert =k_i$ .

In particular, B itself is the only subset of B of grid-complexity $0$ .

Example 2.13. Suppose that V is an ordered vector space over an ordered division ring, $d = d_1 + \dotsb + d_r \in \mathbb {N}$ and

$$ \begin{align*} X = \left\{ \bar{x} \in V^{d}:f_{1}\left(\bar{x}\right) \leq 0, \dotsc, f_{p}\left(\bar{x}\right) \leq 0,f_{p+1}\left(\bar{x}\right)<0,\dotsc,f_{s}\left(\bar{x}\right)<0\right\}, \end{align*} $$

for some linear functions $f_i: V^d \to V, i \in [s]$ . Then each $f_i$ is coordinate-wise monotone (Example 2.3), and hence each of the sets

$$ \begin{align*}\left\{\bar{x} \in V^d : f_i(\bar{x}) <0 \right\}, \left\{\bar{x} \in V^d : f_i(\bar{x}) \leq 0 \right\}\end{align*} $$

is a basic subset of the grid $V^{d_1} {{\times }\dotsb {\times }} V^{d_r}$ (the latter by Remark 2.7), and $X \subseteq V^{d_1} {{\times }\dotsb {\times }} V^{d_r}$ as an intersection of these s basic sets has grid-complexity s.

Remark 2.14.

  1. 1. Let B be an r-grid and $A\subseteq B$ a subset of B of grid-complexity s. If $C \subseteq B$ is a subgrid containing A, then A is also a subset of C of grid-complexity s.

  2. 2. In particular, if $A\subseteq B$ is a subset of grid-complexity s, then A is a subset of grid-complexity $ s$ of the grid $A_1{{\times }\dotsb {\times }} A_r$ , where $A_i :=\pi _i(A)$ is the projection of A on $B_i$ (it is the smallest subgrid of B containing A).

Definition 2.15. Let $B=B_1{{\times }\dotsb {\times }} B_r$ be a finite r-grid and set $n_i :=\lvert B_i\rvert $ . For $j\in \{0,\dotsc , r\}$ , we will denote by $\delta _j^r(B)$ the integer

$$ \begin{align*} \delta_j^r(B) := \sum_{ i_1<i_2<\dotsb< i_j \in [r]} n_{i_1} \cdot n_{i_2} \cdot \dotsb \cdot n_{i_j}.\end{align*} $$

Example 2.16. We have $\delta ^r_0(B)=1$ , $\delta ^r_1(B)=n_1+\dotsb +n_r$ , $\delta _r^r(B)=n_1n_2\dotsb n_r$ .

We can now state the main theorem:

Theorem 2.17. For all integers $r\geq 2, s\geq 0, k\geq 2$ , there are $\alpha =\alpha (r,s,k)\in \mathbb {R}$ and $\beta =\beta (r,s)\in \mathbb N$ such that for any finite r-grid B and $K_{k,\dotsc ,k}$ -free subset $A \subseteq B$ of grid-complexity s, we have

$$ \begin{align*} |A| \leq \alpha \delta^r_{r-1}(B) \log^\beta \left( \delta^r_{r-1}(B)+1 \right). \end{align*} $$

Moreover, we can take $\beta (r,s) := s\left (2^{r-1}-1\right )$ .

Remark 2.18. Inspecting the proof in Sections 2.3 and 2.4, it can be verified that the dependence of $\alpha $ on k in Theorem 2.17 s at most linear.

Remark 2.19. We use $\log ^\beta \left ( \delta ^r_{r-1}(B)+1 \right )$ instead of $\log ^\beta \left ( \delta ^r_{r-1}(B) \right )$ to include the case $\delta ^r_{r-1}(B)\leq 1$ .

Remark 2.20. If, in Theorem 2.17, A is only assumed to be a union of at most t sets of grid-complexity s, then the same bound holds with $\alpha ' := t \cdot \alpha $ (if $A = \bigcup _{i \in [t]} A_i$ is $K_{k,\dotsc ,k}$ -free, then each $A_i$ is also $K_{k,\dotsc ,k}$ -free, so we can apply Theorem 2.17 to each $A_i$ and bound $\lvert A\rvert $ by the sum of their bounds).

Definition 2.21. Let $B=B_1{{\times }\dotsb {\times }} B_r$ be a grid. We extend the definition of $\delta ^r_j$ to arbitrary finite subsets of B as follows: let $A\subseteq B$ be a finite subset, and let $A_i :=\pi _i(A)$ , $i\in [r]$ , be the projections of A. We define $\delta ^r_j(A) :=\delta ^r_j(A_1{{\times }\dotsb {\times }} A_r)$ .

If B is a finite r-grid and $A\subseteq B$ , then obviously $\delta ^r_j(A)\leq \delta ^r_j(B)$ . Thus Theorem 2.17 is equivalent to the following:

Proposition 2.22. For all integers $r\geq 2, s\geq 0, k\geq 2$ , there are $\alpha =\alpha (r,s,k)\in \mathbb {R}$ and $\beta =s\left (2^{r-1}-1\right ) \in \mathbb N$ such that for any r-grid B and $K_{k,\dotsc ,k}$ -free finite subset $A \subseteq B$ of grid-complexity $\leq s$ , we have

$$ \begin{align*} \lvert A\rvert \leq \alpha \delta^r_{r-1}(A) \log^\beta\left(\delta^r_{r-1}(A)+1\right). \end{align*} $$

Definition 2.23. For $r\geq 1, s\geq 0, k\geq 2$ and $n\in \mathbb N$ , let $F_{r,k}(s,n)$ be the maximal size of a $K_{k,\dotsc ,k}$ -free subset A of grid-complexity s of some r-grid B with $\delta _{r-1}^r(B)\leq n$ .

Then Proposition 2.22 can be restated as follows:

Proposition 2.24. For all integers $r\geq 2, s\geq 0, k\geq 2$ , there are $\alpha =\alpha (r,s,k)\in \mathbb {R}$ and $\beta =\beta (r,s)\in \mathbb N$ such that

$$ \begin{align*} F_{r,k}(s,n) \leq \alpha n \log^\beta( n+1).\end{align*} $$

Remark 2.25. Notice that $F_{r,k}(s,0)=0$ .

In the rest of the section we prove Proposition 2.24 by induction on r, where for each r it is proved by induction on s. We will use the following simple recurrence bound:

Fact 2.26. Let $\mu \colon \mathbb N \to \mathbb N$ be a function satisfying $\mu (0)=0$ and $\mu (n)\leq 2\mu (\lfloor n/2\rfloor ) + \alpha n \log ^\beta (n+1)$ for some $\alpha \in \mathbb {R}$ and $\beta \in \mathbb N$ . Then $\mu (n)\leq \alpha ' n \log ^{\beta +1} (n+1)$ for some $\alpha '=\alpha '(\alpha ,\beta )\in \mathbb {R}$ .

2.3. The base case $r=2$

Let $B=B_1{\times } B_2$ be a finite grid and $A\subseteq B$ a subset of grid-complexity s. We will proceed by induction on s.

If $s=0$ , then $A=B_1\times B_2$ . If A is $K_{k,k}$ -free, then one of the sets $B_1, B_2$ must have size at most k. Hence $\lvert A\rvert \leq k(\lvert B_1\rvert +\lvert B_2\rvert )=k\delta ^2_1(B)$ .

Thus

$$ \begin{align*} F_{2,k}(0,n) \leq k n.\end{align*} $$

Remark 2.27. The same argument shows that $F_{r,k}(0,n) \leq k n$ for all $r \geq 2$ .

Assume now that the theorem is proved for $r=2$ and all $s'<s$ . Define $n_1 :=\lvert B_1\rvert $ , $n_2 :=\lvert B_2\rvert $ and $n :=\delta ^2_1(B)=n_1+n_2$ .

We choose basic sets $X_1,\dotsc , X_s \subseteq B$ such that $A=B \cap \bigcap _{j \in [s]} X_j$ .

By Proposition 2.8, we can choose a finite linear order $(S,<)$ and functions $f_1\colon B_1\to S$ and $f_2\colon B_2\to S$ so that

$$ \begin{align*} X_s=\left\{ (x_1,x_2) \in B_1\times B_2 \colon f_1(x_1)< f_2(x_2)\right\}.\end{align*} $$

For $l\in S$ , $i\in \{1,2\}$ and $\square \in \{ <,=,>, \leq , \geq \}$ , let

$$ \begin{align*} B_i^{\square l} = \left\{ b\in B_i \colon f_i(b) \square l \right\}. \end{align*} $$

We choose $h\in S$ such that

$$ \begin{align*} \left\lvert B_1^{<h}\right\rvert+\left\lvert B_2^{<h}\right\rvert \leq n/2 \text{ and } \left\lvert B_1^{>h}\right\rvert+\left\lvert B_2^{>h}\right\rvert \leq n/2. \end{align*} $$

For example, we can take h to be the minimal element in $f_1(B_1)\cup f_2(B_2)$ with $ \left \lvert B_1^{\leq h}\right \rvert +\left \lvert B_2^{ \leq h}\right \rvert \geq n/2$ . Then

$$ \begin{align*} X_s = \left[\left(B_1^{<h}\times B_2^{<h}\right) \cap X_s \right] \cup \left[ \left(B_1^{>h}\times B_2^{>h} \right) \cap X_s \right] \cup \left(B_1^{<h}\times B_2^{\geq h} \right) \cup \left(B_1^{=h}\times B_2^{>h}\right). \end{align*} $$

Hence we conclude

$$ \begin{align*}F_{2,k}(s,n)\leq 2 F_{2,k}(s,\lfloor n/2\rfloor)+2F_{2,k}(s-1,n).\end{align*} $$

Applying the induction hypothesis on s and using Fact 2.26 and Remark 2.25, we obtain $F_{2,k}(s,n)\leq \alpha n (\log n)^\beta $ for some $\alpha =\alpha (s,k)\in \mathbb {R}$ and $\beta =\beta (s)\in \mathbb N$ .

This finishes the base case $r=2$ .

2.4. Induction step

We fix $r \in \mathbb {N}_{\geq 3}$ and assume that Proposition 2.24 holds for all pairs $(r',s)$ with $r'<r$ and $s \in \mathbb {N}$ .

Definition 2.28. Let $B=B_1{{\times }\dotsb {\times }} B_r$ be a finite r-grid.

  1. 1. For integers $t, u\in \mathbb N$ , we say that a subset $A\subseteq B$ is of split grid-complexity $(t, u)$ if there are basic sets $X_1,\dotsc , X_{u} \subseteq B$ , a subset $A^r\subseteq B_1{{\times }\dotsb {\times }} B_{r-1}$ of grid-complexity t and a subset $A_r\subseteq B_{r}$ such that $A=(A^r\times A_r)\cap \bigcap _{i \in [u]} X_i$ .

  2. 2. For $t, u \geq 0, k\geq 2$ and $n\in \mathbb N$ , let $G_{k}(t,u,n)$ be the maximal size of a $K_{k,\dotsc ,k}$ -free subset A of an r-grid B of split grid-complexity $(t,u)$ with $\delta _{r-1}^r(B)\leq n$ .

Remark 2.29.

  1. 1. Note that $A_r$ has grid-complexity at most $1$ , which is the reason we do not include a parameter for the grid-complexity of $A_r$ in the split grid-complexity of A.

  2. 2. If $A\subseteq B$ is of grid-complexity s, then it is of split grid-complexity $(0,s)$ .

  3. 3. If $A\subseteq B$ is of split grid-complexity $(t, u)$ , then it is of grid-complexity $t + u$ .

For the rest of the proof, we abuse notation slightly and refer to the split grid-complexity of a set as simply the grid-complexity. To complete the induction step we will prove the following proposition:

Proposition 2.30. For any integers $t,u\geq 0, k\geq 2, r \geq 3$ , there are $\alpha ' = \alpha '(r,k,t,u)\in \mathbb {R}$ and $\beta ' = \beta '(r,k,t,u)\in \mathbb N$ such that

$$ \begin{align*} G_{k}(t,u,n) \leq \alpha' n \log^{\beta'}(n+1).\end{align*} $$

We will use the following notations throughout the section:

  • $B=B_1{{\times }\dotsb {\times }} B_r$ is a finite grid with $n=\delta ^r_{r-1}(B)$ ;

  • $A\subseteq B$ is a subset of grid-complexity $(t,u)$ ;

  • $B^r$ is the $(r-1)$ -grid $B^r :=B_1{{\times }\dotsb {\times }} B_{r-1}$ ;

  • $A^r \subseteq B^r$ is a subset of grid-complexity t, $A_r \subseteq B_r$ , and $X_1,\dotsc X_{u} \subseteq B$ are basic subsets such that $A= (A^r{\times } A_r)\cap \bigcap _{i \in [u]} X_i$ .

We proceed by induction on u.

The base case $u=0$ of Proposition 2.30.

In this case, $A=A^r\times A_r$ . If A is $K_{k,\dotsc ,k}$ -free, then either $A^r$ is $K_{k,\dotsc ,k}$ -free or $\lvert A_r\rvert <k$ .

In the first case, by the induction hypothesis on r, there are $\alpha =\alpha (r-1, t,k)$ and $\beta =\beta (r-1, t)$ such that $\lvert A^r\rvert \leq \alpha \delta ^{r-1}_{r-2}(B^r)\log ^\beta \left ( \delta ^{r-1}_{r-2}(B^r)+1\right )$ . In the second case, we have $\lvert A\rvert \leq \lvert B^r\rvert k=\delta ^{r-1}_{r-1}(B^r)k$ .

Since $n=\delta ^r_{r-1}(B)=\delta ^{r-1}_{r-1}(B^r)+\delta ^{r-1}_{r-2}(B^r) \lvert B_r\rvert $ , the conclusion of the proposition follows with $\alpha ' := \alpha , \beta ' := \beta $ .

Induction step of Proposition 2.30.

We assume now that the proposition holds for all pairs $(t,u')$ with $u'<u$ and $t \in \mathbb {N}$ .

Given a tuple $x = (x_1, \dotsc , x_r) \in B$ , we set $x^r := (x_1, \dotsc , x_{r-1})$ . By Proposition 2.8, we can choose a finite linear order $(S,<)$ , a coordinate-wise monotone function $f^r\colon B^r\to S$ and a function $f_r\colon B_r\to S$ so that

$$ \begin{align*} X_u=\left\{ x^r \oplus_r x_r \in B^r\times B_r \colon f^r(x^r)< f_r(x_r)\right\}.\end{align*} $$

Moreover, by Remark 2.9 we may assume without loss of generality that the coordinate-wise monotone function defining $X_u$ is given by

$$ \begin{align*} f(x^r \oplus_r x_r)=f^r(x^r) - f_r(x_r). \end{align*} $$

Definition 2.31. Given an arbitrary set $C^r \subseteq B^r$ , we say that a set $H^r \subseteq C^r$ is an $f^r$ -strip in $C^r$ if

$$ \begin{align*} H^r=\left\{ x^r \in C^r \colon l_1 \triangleleft_1 f^r(x^r) \triangleleft_2 l_2\right\}\end{align*} $$

for some $l_1,l_2\in S$ , $\triangleleft _1, \triangleleft _2\in \{ <,\leq \}$ . Likewise, given an arbitrary set $C_r \subseteq B_r$ , we say that $H_r \subseteq C_r$ is an $f_r$ -strip in $C_r$ if

$$ \begin{align*} H_r=\left\{ x_r \in C_r \colon l_1 \triangleleft_1 f_r(x_r) \triangleleft_2 l_2\right\}\end{align*} $$

for some $l_1,l_2\in S$ , $\triangleleft _1, \triangleleft _2\in \{ <,\leq \}$ . If $C^r = A^r$ or $C_r = A_r$ , we simply say an $f^r$ -strip or $f_r$ -strip, respectively.

Remark 2.32. Note the following:

  1. 1. $A^r$ is an $f^r$ -strip, and $A_r$ is an $f_r$ -strip.

  2. 2. Every $f^r$ -strip is a subset of the $(r-1)$ -grid $B^r$ of grid-complexity $t+2$ (using Remark 2.7).

  3. 3. The intersection of any two $f^r$ -strips is an $f^r$ -strip; the same conclusion holds for $f_r$ -strips.

Definition 2.33.

  1. 1. We say that a subset $H\subseteq B$ is an f-grid if $H=H^r\times H_r$ , where $H^r\subseteq B^r$ is an $f^r$ -strip in $B^r$ and $H_r \subseteq B_r$ is an $f_r$ -strip in $B_r$ .

  2. 2. If $H=H^r\times H_r$ is an f-grid, we set

    $$ \begin{align*} \Delta(H) :=\lvert H^r\rvert + \delta^{r-1}_{r-2}(H^r)\lvert H_r\rvert \text{ (see Definition~2.21 for } \delta^{r-1}_{r-2}). \end{align*} $$
    Note that if H is a sub-grid of B, then $\Delta (H)=\delta ^r_{r-1}(H)$ .
  3. 3. For an f-grid H, we will denote by $A_H$ the set $A\cap H$ .

The induction step for Proposition 2.30 will follow from the following proposition:

Proposition 2.34. For all integers $k\geq 2, r \geq 3$ , there are $\alpha ' = \alpha '(r,k,t,u)\in \mathbb {R}$ and $\beta '=\beta '(r,t,u) \in \mathbb N$ such that, for any f-grid H, if the set $A_H$ is $K_{k,\dotsc ,k}$ -free then

$$ \begin{align*} \lvert A_H\rvert \leq \alpha' \Delta(H)\log^{\beta'}(\Delta(H)+1). \end{align*} $$

We should stress that in this proposition, $\alpha '$ and $\beta '$ do not depend on $f^r, f_r$ , B, $A^r$ , and $A_r$ , but they may depend on our fixed t and u.

Given Proposition 2.34, we can apply it to the f-grid $H := A^r\times A_r$ (so $A_H = A$ ) and get

$$ \begin{align*}\lvert A\rvert\leq \alpha' \Delta(H)\log^{\beta'}(\Delta(H)+1).\end{align*} $$

It is easy to see that $\Delta (A^r\times A_r)\leq \delta ^r_{r-1}(B)$ , and hence Proposition 2.30 follows with the same $\alpha '$ and $\beta '$ .

We proceed with the proof of Proposition 2.34:

Proof of Proposition 2.34. Fix $m\in \mathbb N$ , and let $L(m)$ be the maximal size of a $K_{k,\dotsc ,k}$ -free set $A_H$ among all f-grids $H \subseteq B$ with $\Delta (H)\leq m$ . We need to show that for some $\alpha '=\alpha '(k)\in \mathbb {R}$ and $\beta ' \in \mathbb N$ we have

$$ \begin{align*}L(m)\leq \alpha' m \log^{\beta'}(m+1).\end{align*} $$

Let $H=H^r\times H_r$ be an f-grid with $\Delta (H)\leq m$ .

For $l\in S$ and $\square \in \{ <,=,>, \leq , \geq \}$ , define

$$ \begin{align*} H^{r,\square l} := \left\{ x^r \in H^r \colon f^r(x^r) \square l \right\} \end{align*} $$

and

$$ \begin{align*} H_r^{\square l} := \left\{ x_r \in H_r \colon f_r(x_r) \square l \right\}. \end{align*} $$

Note that for every $l \in S$ , $H^{r,\square l}$ is an $f^r$ -strip in $H^r$ , $H_r^{\square l}$ is an $f_r$ -strip in $H_r$ and their product is an f-grid.

Claim 2.35. There is $h\in S$ such that

$$ \begin{align*} \Delta\left(H^{r,<h}\times H_r^{<h}\right) \leq m/2 \text{ and } \Delta\left(H^{r,>h}\times H_r^{>h}\right) \leq m/2.\end{align*} $$

Proof. Set $\delta :=\delta ^{r-1}_{r-2}(H^r)$ .

Let h be the minimal element in $f^r(H^r)\cup f_r(H_r)$ with

$$ \begin{align*} \left\lvert H^{r,\leq h }\right\rvert+\delta \left\lvert H_r^{\leq h}\right\rvert \geq m/2.\end{align*} $$

Then $ \left \lvert H^{r,< h }\right \rvert +\delta \left \lvert H_r^{< h}\right \rvert \leq m/2$ and $ \left \lvert H^{r,> h }\right \rvert +\delta \left \lvert H_r^{> h}\right \rvert \leq m/2$ . Since $H^{r,< h}, H^{r,> h} \subseteq H^r$ , we have $\delta ^{r-1}_{r-2}\left (H^{r,< h}\right ), \delta ^{r-1}_{r-2}\left (H^{r,> h}\right ) \leq \delta $ . The claim follows.

Let h be as in the claim. It is not hard to see that the following hold:

$$ \begin{gather*} \left( H^{r, \leq h} \times H_r^{\geq h} \right) \cap X_u = \left( H^{r, < h} \times H_{r}^{\geq h} \right) \cup \left( H^{r, =h} \times H_r^{>h} \right),\\ \left( H^{r, \geq h} \times H_r^{\leq h} \right) \cap X_u = \emptyset. \end{gather*} $$

It follows that

$$ \begin{align*} A_H \cap X_u = \left[\left(H^{r,<h}\times H_r^{<h}\right) \cap X_u \right] \cup \left[\left(H^{r,>h}\times H_r^{>h}\right) \cap X_u \right] \cup \left(H^{r,<h}\times H_r^{\geq h}\right) \cup \left(H^{r,=h}\times H_r^{>h}\right). \end{align*} $$

Hence, by the choice of h and using Remark 2.32(2),

$$ \begin{align*}L(m)\leq 2 L(\lfloor m/2\rfloor)+ 2G_{k}(t+2,u-1,m).\end{align*} $$

Applying the induction hypothesis on u and using Fact 2.26, we obtain $L(m)\leq \alpha ' m \log ^{\beta '}(m+1)$ for some $\alpha '=\alpha '(k)\in \mathbb {R}$ and $\beta '\in \mathbb N$ .

This finishes the proof of Proposition 2.34, and hence of the induction step of Proposition 2.24.

Finally, inspecting the proof, we have shown the following:

  1. 1. $\beta (2,s) \leq s$ for all $s \in \mathbb {N}$ .

  2. 2. $\beta '(r,t, 0) \leq \beta (r-1,t)$ for all $r \geq 3$ and $t \in \mathbb {N}$ .

  3. 3. $\beta '(r,t,u) \leq \beta '(r, t+2, u-1) + 1$ for all $r \geq 3, t \geq 0, u \geq 1$ .

Iterating (3), for every $r \geq 3, s \geq 1$ we have $\beta (r,s) \leq \beta '(r,0,s) \leq \beta '(r, 2s, 0) + s$ . Hence, by (2), $\beta (r,s) \leq \beta (r-1, 2s) + s$ for every $r \geq 3$ and $s \geq 1$ . Iterating this, we get $\beta (r,s) \leq \beta \left (2, 2^{r-2}s\right ) + s \sum _{i=0}^{r-3}2^i$ . Using (1), this implies $\beta (r,s) \leq s \sum _{i=0}^{r-2}2^i = s\left (2^{r-1}-1\right )$ for all $r \geq 3, s \geq 1$ . Hence, by Remark 2.27 and (1) again, $\beta (r,s) \leq s\left (2^{r-1}-1\right )$ for all $r \geq 2, s \geq 0$ .

2.5. Some applications

We observe several immediate applications of Theorem 2.17, starting with the following bound for semilinear hypergraphs:

Corollary 2.36. For every $r,s,t,k \in \mathbb {N}, r \geq 2$ , there exist some $\alpha =\alpha (r,s,t,k)\in \mathbb {R}$ and $\beta (r,s) := s\left (2^{r-1}-1\right )$ satisfying the following: for any semilinear $K_{k, \dotsc , k}$ -free r-hypergraph $H = (V_1, \dotsc , V_r;E)$ of description complexity $(s,t)$ (see Definition 1.7), taking $V:= \prod _{i \in [r]}V_i$ we have

$$ \begin{align*} \lvert E\rvert \leq \alpha \delta^r_{r-1}(V) \log^\beta \left( \delta^r_{r-1}(V)+1 \right). \end{align*} $$

Proof. By assumption, the edge relation E can be defined by a union of t sets, each of which is defined by s linear equalities and inequalities, hence of grid-complexity $\leq s$ (see Example 2.13). The conclusion follows by Theorem 2.17 and Remark 2.20.

As a special case with $r=2$ , this implies a bound for the following incidence problem:

Corollary 2.37. For every $s,k \in \mathbb {N}$ there exists some $\alpha =\alpha (s,k)\in \mathbb {R}$ satisfying the following:

Let $d \in \mathbb {N}$ and $H_1, \dotsc , H_s \subseteq \mathbb {R}^d$ be finitely many (closed or open) half-spaces in $\mathbb {R}^d$ . Let $\mathcal {F}$ be the (infinite) family of all possible polytopes in $\mathbb {R}^d$ cut out by arbitrary translates of $H_1, \dotsc , H_s$ .

For any set P of $n_1$ points in $\mathbb {R}^d$ and any set F of $n_2$ polytopes in $\mathcal {F}$ , if the incidence graph on $P \times F$ is $K_{k,k}$ -free, then it contains at most $\alpha n \log ^{s} n$ incidences.

Proof. We can write

$$ \begin{align*}H_i = \left\{\bar{x} = (x_1, \dotsc, x_d) \in \mathbb{R}^d : \sum_{j \in [d]} a_{i,j} x_j \square_i b_i \right\},\end{align*} $$

where $a_{i,j},b_i \in \mathbb {R}$ and $\square _i \in \{>, \geq \}$ for $i \in [s], j \in [d]$ depending on whether $H_i$ is an open or a closed half-space.

Every polytope $F \in \mathcal {F}$ is of the form $\bigcap _{i \in [s]} (\bar {y}_i + H_i)$ for some $(\bar {y}_1, \dotsc , \bar {y}_s) \in \mathbb {R}^{sd}$ , where $\bar {y}_i + H_i$ is the translate of $H_i$ by the vector $\bar {y}_i = \left (y_{i,1}, \dotsc , y_{i,d}\right ) \in \mathbb {R}^d$ – that is,

$$ \begin{align*}\bar{y}_i + H_i = \left\{ \bar{x} \in \mathbb{R}^d : \sum_{j \in [d]} a_{i,j} x_j + \sum_{j \in [d]}\left(-a_{i,j}\right)y_j \square_i b_i \right\}.\end{align*} $$

Then the incidence relation between points in $\mathbb {R}^d$ and polytopes in $\mathcal {F}$ can be identified with the semilinear set

$$ \begin{align*}\left\{ \left(\bar{x}; \left(y_{i,j}\right)_{i \in [s], j \in [d]} \right) \in \mathbb{R}^d \times \mathbb{R}^{sd} : \bigwedge_{i \in [s]} \sum_{j \in [d]} a_{i,j} x_j + \sum_{j \in [d]}\left(-a_{i,j}\right)y_{i,j} \square_i b_i \right\}\end{align*} $$

defined by s linear inequalities. The conclusion now follows by Corollary 2.36 with $r=2$ .

In particular, we get a bound for the original question that motivated this paper.

Corollary 2.38. Let $\mathcal {F}_d$ be the family of all (closed or open) boxes in $\mathbb {R}^d$ . Then for every k there exists some $\alpha = \alpha (d,k)$ satisfying the following: for any set P of $n_1$ points in $\mathbb {R}^d$ and any set F of $n_2$ boxes in $\mathcal {F}_d$ , if the incidence graph on $P \times F$ is $K_{k,k}$ -free, then it contains at most $\alpha n \log ^{2 d} n $ incidences.

Proof. This is immediate from Corollary 2.37, since we have $2d$ half-spaces in $\mathbb {R}^d$ such that every box in $\mathbb {R}^d$ is cut out by the intersection of their translates.

3. Lower bounds

While we do not know if the bound $\beta (2,s) \leq s$ in Theorem 2.17 is optimal, in this section we show that at least the logarithmic factor is unavoidable already for the incidence relation between points and dyadic boxes in $\mathbb {R}^2$ .

We describe a slightly more general construction first. Fix $d \in \mathbb {N}_{>0}$ .

Definition 3.1. Given finite tuples $\bar {p}=(p_1, \dotsc , p_{n}), \bar {q}=(q_1, \dotsc ,q_n)$ and $\bar {r}=(r_1, \dotsc , r_m)$ with $p_i,q_i,r_i \in \mathbb {R}^d$ – say $p_i = \left (p_{i,1}, \dotsc , p_{i,d}\right ), q_i = \left (q_{i,1}, \dotsc , q_{i,d}\right ), r_i = \left (r_{i,1}, \dotsc , r_{i,d}\right )$ – we say that $\bar {p}$ and $\bar {q}$ have the same order-type over $\bar {r}$ if

$$ \begin{gather*} p_{i,j} \square p_{i',j'} \iff q_{i,j} \square q_{i',j'} \text{ and} \\ p_{i,j} \square r_{k,j'} \iff q_{i,j} \square r_{k,j'} \end{gather*} $$

for all $\square \in \{<,>,= \}$ , $1 \leq i,i' \leq n, 1 \leq j,j' \leq d$ and $1 \leq k \leq m$ .

In other words, the tuples $\left (p_{i,j} : 1 \leq i \leq n, 1\leq j \leq d\right )$ and $\left (q_{i,j} : 1 \leq i \leq n, 1\leq j \leq d\right )$ have the same quantifier-free type over the set $\left \{r_{i,j} : 1 \leq i \leq m, 1 \leq j \leq d \right \}$ in the structure $(\mathbb {R}, <)$ .

Remark 3.2. Assume that $P =\{ p_1, \dotsc , p_{n} \} \subseteq \mathbb {R}^d$ is a finite set of points and B is a finite set of d-dimensional open boxes with axis-parallel sides, with I incidences between P and B.

  1. 1. By perturbing P and B slightly, we may assume that for every $1 \leq j \leq d$ , all points in P have pairwise distinct jth coordinates $p_{1,j}, \dotsc , p_{n,j}$ , and none of the points in P belongs to the border of any of the boxes in B, while the incidence graph between P and B remains unchanged.

  2. 2. Let $\bar {r}$ be the tuple listing all corners of all boxes in B. If $P' = \left \{p^{\prime }_1, \dotsc , p^{\prime }_n\right \} \subseteq \mathbb {R}^d$ is an arbitrary set of points with the same order-type as P over $\bar {r}$ , then the incidence graph on $P \times B$ is isomorphic to the incidence graph on $P' \times B$ .

We have the following lemma for combining point-box incidence configurations in a higher-dimensional space:

Lemma 3.3. Given any $d,n_1,n_2,n^{\prime }_1, n^{\prime }_2, m,m' \in \mathbb {N}_{>0}$ , assume that:

  1. 1. there exists a set of points $P^{d-1} \subseteq \mathbb {R}^{d-1}$ with $\left \lvert P^{d-1}\right \rvert = n_1$ and a set of $(d-1)$ -dimensional boxes $B^{d-1}$ with $\left \lvert B^{d-1}\right \rvert = n_2$ , with m incidences between them and the incidence graph $K_{2,2}$ -free; and

  2. 2. there exists a set of points $P^d \subseteq \mathbb {R}^d$ with $\left \lvert P^d\right \rvert = n^{\prime }_1$ and a set of d-dimensional boxes $B^d$ with $\left \lvert B^d\right \rvert =n^{\prime }_2$ , with $m'$ incidences between them and the incidence graph $K_{2,2}$ -free.

Then there exists a set of points $P \subseteq \mathbb {R}^d$ with $\lvert P\rvert = n_1 n^{\prime }_1$ and a set of d-dimensional boxes B with $\lvert B\rvert = n_1n^{\prime }_2+n^{\prime }_1n_2$ , so that there are $n_1m' + m n^{\prime }_1$ incidences between P and B and their incidence graph is still $K_{2,2}$ -free.

Proof. By Remark 3.2(1) we may assume that for every $1 \leq j \leq d$ , all points in $P^d$ have pairwise distinct jth coordinates; for every $1 \leq j \leq d-1$ , all points in $P^{d-1}$ have pairwise distinct jth coordinates; and none of the points is on the border of any of the boxes. Write $P^{d-1}$ as $p_1, \dotsc , p_{n_1}$ . Let $\bar {r}$ be the tuple listing all corners of all boxes in $B^{d-1}$ .

Using this, for each $p_i$ we can choose a very small $(d-1)$ -dimensional box $\beta _i$ with $p_i \in \beta _{i}$ and such that for any choice of points $p^{\prime }_i \in \beta _i, 1 \leq i \leq n_1$ , we have that $\left (p^{\prime }_1, \dotsc , p^{\prime }_{n_1}\right )$ has the same order-type as $(p_1, \dotsc , p_{n_1})$ over $\bar {r}$ . In particular, every $\beta _i$ is pairwise disjoint, and the incidence graph between $P^{d-1}$ and $B^{d-1}$ is isomorphic to the incidence graph between $\left (p^{\prime }_i, \dotsc , p^{\prime }_{n_1}\right )$ and $B^{d-1}$ by Remark 3.2(2).

Contracting and translating while keeping the dth coordinate unchanged, for each $1 \leq i \leq n_1$ we can find a copy $\left (P^{d}_i, B^d_i\right )$ of the configuration $\left (P^d, B^d\right )$ entirely contained in the box $\beta _i \times \mathbb {R}$ – that is,

  • all points in $P^{d}_i$ and boxes in $ B^d_i$ are contained in $\beta _i \times \mathbb {R}$ ;

  • the incidence graph on $\left (P^{d}_i, B^d_i\right )$ is isomorphic to the incidence graph on $\left (P^{d}, B^d\right )$ ; and

  • for all i, the dth coordinate of every point in $P_i^d$ is the same as the dth coordinate of the corresponding point in $P^d$ .

Set $P := \bigcup _{1 \leq i \leq n_1} P^d_i$ and $B' := \bigcup _{1 \leq i \leq n_1} B^d_i$ ; then $\lvert P\rvert = n_1n^{\prime }_1, \lvert B'\rvert = n_1n^{\prime }_2$ and there are $n_1m'$ incidences between P and $B'$ .

Write $P^d$ as $q_1, \dotsc , q_{n^{\prime }_1}$ and $B^{d-1}$ as $c_1, \dotsc , c_{n_2}$ . As all of the dth coordinates of the points in $P^d$ are pairwise disjoint, for each $1 \leq j \leq n^{\prime }_1$ we can choose a small interval $I_j \subseteq \mathbb {R}$ with $q_{j,d} \in I_j$ and such that all of the intervals $I_j, 1 \leq j \leq n^{\prime }_1$ , are pairwise disjoint. For each $1 \leq j \leq n^{\prime }_1$ and $c_l \in B^{d-1}$ , we consider the d-dimensional box $c_{j,l} :=c_l \times I_j$ . Define $B_j := \left \{c_{j,l} : 1 \leq l \leq n_2 \right \}$ . For each $1 \leq i \leq n_1$ and $1 \leq j \leq n^{\prime }_1$ , $(\beta _i \times \mathbb {R}) \cap \left (\mathbb {R}^{d-1} \times I_j\right )$ contains exactly one point $q_{i,j}$ (given by the copy of $q_{j}$ in $P_i^d$ ), and the projection $q^{\prime }_{i,j}$ of $q_{i,j}$ onto the first $d-1$ coordinates is in $\beta _i$ . Hence the incidence graph between P and $B_j$ is isomorphic to the incidence graph between $P^{d-1}$ and $B^{d-1}$ by the choice of the $\beta _i$ s, and in particular the number of incidences is m.

Finally, define $B := B' \cup \bigcup _{1 \leq j \leq n^{\prime }_1} B_j$ ; then $\lvert B\rvert = n_1n^{\prime }_2 + n^{\prime }_1n_2$ . Note that $c_{j,l} \cap c_{j',l'} = \emptyset $ for $j \neq j'$ and any $l,l'$ – that is, no box in $B_j$ intersects any of the boxes in $B_{j'}$ for $j\neq j'$ . It is now not hard to check that the incidence graph between P and B is $K_{2,2}$ -free, by construction and the assumptions of $K_{2,2}$ -freeness of $\left (P^d,B^d\right )$ and $\left (P^{d-1}, B^{d-1}\right )$ , and that there are $n_1m'+mn^{\prime }_1$ incidences between P and B.

Remark 3.4. It follows from the proof that if all the boxes in $B^{d-1}$ and $B^d$ are dyadic (see Definition 4.6), then we can choose the boxes in B to be dyadic as well.

Proposition 3.5. For any $\ell \in \mathbb {N}$ , there exist a set P of $ \ell ^{\ell }$ points and a set B of $\ell ^{\ell }$ dyadic boxes in $\mathbb {R}^2$ such that their incidence graph is $K_{2,2}$ -free and the number of incidences is $\ell \ell ^{\ell }$ .

In particular, substituting $n := \ell ^{\ell }$ , this shows that the number of incidences grows as $\Omega \left (n \frac {\log n}{\log \log n} \right )$ .

Proof. Given d, assume that there exist $K_{2,2}$ -free ‘point–dyadic box’ configurations satisfying Lemma 3.3(1) and (2) for some parameters $d, n_1, n_2, n^{\prime }_1, n^{\prime }_2, m, m'$ . Then for any $j \in \mathbb {N}$ , we can iterate the lemma j times and find a $K_{2,2}$ -free ‘point–dyadic box’ configuration in $\mathbb {R}^d$ with $n_1^j n^{\prime }_1$ points, $n_1^j n^{\prime }_2 + j n_1^{j-1} n^{\prime }_1 n_2$ dyadic boxes (Remark 3.4) and $n_1^j m' + j n_1^{j-1} n^{\prime }_1 m$ incidences.

In particular, let $d = 2$ and let $\ell $ be arbitrary. We can start with $n_1 = \ell , n_2 = 1, m=\ell $ (one dyadic interval containing $n_1$ points in $\mathbb {R}$ ) and $n^{\prime }_1=1, n^{\prime }_2=0, m' = 0$ (one point and zero dyadic boxes in $\mathbb {R}^2$ ). Taking $j := \ell $ , we then find a $K_{2,2}$ -free configuration with $\ell ^{\ell }$ points, $\ell ^\ell $ dyadic boxes and $\ell \ell ^{\ell }$ incidences. Hence for $n := k^k$ , we have n points, n boxes and $\Omega \left (n \frac {\log n}{\log \log n} \right )$ incidences.

Remark 3.6. We remark that the construction in Lemma 3.3 cannot produce a $K_{2,2}$ -free configuration with more than $O \left ( n \frac {\log n}{\log \log n} \right )$ incidences in $\mathbb {R}^d$ for any d.

Indeed, using the ‘coordinates’ $\left ( \log n^{\prime }_1, \frac {n^{\prime }_2}{n^{\prime }_1}, \frac {m'}{n^{\prime }_1} \right )$ instead of $\left (n^{\prime }_1,n^{\prime }_2,m'\right )$ , where the coordinates correspond to the number of points, boxes and incidences, respectively, the lemma says that if $\left ( \log n_1, \frac {n_2}{n_1}, \frac {m}{n_1} \right )$ is attainable in $d-1$ dimensions and $\left ( \log n^{\prime }_1, \frac {n^{\prime }_2}{n^{\prime }_1}, \frac {m'}{n^{\prime }_1} \right )$ is attainable in d dimensions, then $\left ( \log n^{\prime }_1 + \log n_1, \frac {n^{\prime }_2}{n^{\prime }_1} + \frac {n_2}{n_1}, \frac {m'}{n^{\prime }_1} + \frac {m}{n_1} \right )$ is attainable in d dimensions. Thus, one adds the vector $\left (\frac {n_2}{n_1}, \frac {m}{n_1} \right )$ to $\left ( \frac {n^{\prime }_2}{n^{\prime }_1},\frac {m'}{n^{\prime }_1} \right )$ . We want to maximise the second coordinate of this vector while keeping the first coordinate below $1$ , and the optimal way to do this essentially is to add $n_1$ times the vector $\left (\frac {1}{n_1},1 \right )$ , which increases $\log n^{\prime }_1$ by $n_1 \log n_1$ and gives the $\frac {\log n}{\log \log n}$ lower bound.

We thus ask whether in the ‘point-box’ incidence bound in $\mathbb {R}^d$ the power of $\log n$ has to grow with the dimension d (see Problem 1.3).

4. Dyadic rectangles

In this section we strengthen the bound on the number of incidences with rectangles on the plane with axis-parallel sides given by Corollary 2.38 – that is, $O_{k} \left ( n \log ^{4} n \right )$ – in the special case of dyadic rectangles, using a different argument (which relies on a certain partial order specific to the dyadic case).

4.1. Locally d-linear orders

Throughout this section, let $(P, \leq )$ be a partially ordered set of size at most $n_1$ , and let L be a collection of subsets of P (possibly with repetitions) of size at most $n_2$ . As before, we let $n = n_1 + n_2$ .

Definition 4.1. We say that a set $S \subseteq P$ is d-linear if it contains no antichains of size greater than d, and $(P, \leq )$ is locally d-linear if any interval $[a,b] = \{ x \in P : a \leq x \leq b \}$ is d-linear.

Note that d-linearity is preserved under removing points from P.

Definition 4.2. The collection L is said to be a $K_{k,k}$ -free arrangement if for any $a_1 \neq \dotsb \neq a_k \in P$ , there are at most $k-1$ sets from L containing all of them simultaneously.

Observe that if one removes any number of points from P or removes any number of sets from L, one still obtains a $K_{k,k}$ -free arrangement. We now state the main theorem of this section:

Theorem 4.3. Suppose $(P,<)$ is locally d-linear and L is a $K_{k,k}$ -free arrangement of d-linear subsets of P. Then

$$ \begin{align*}\sum_{\ell \in L} \lvert\ell\rvert = O_{d,k}\left( n \frac{\log(100+n_1)}{\log \log(100+n_1)} \right).\end{align*} $$

To prove this theorem, we first need some definitions and a lemma. If $x \in P$ , define a parent of x to be an element $y \in P$ with $y>x$ and no element between x and y, and similarly define a child of x to be an element $z \in P$ with $z<x$ and no element between z and x. We say that z is a strict t-descendant of x if there are some elements $z_0 = x> z_1 > \dotsb > z_{t} = z$ such that $z_{i+1}$ is a child of $z_i$ , and that z is a t-descendant of x if it is a strict s-descendant for some $0 \leq s \leq t$ .

Lemma 4.4. Fix $d,k \in \mathbb {N}$ . Let L be a $K_{k,k}$ -free arrangement of d-linear subsets of P and let $m> 0$ . Let $P'$ denote the set of all elements in P which have a $(k-1)$ -descendant with more than m children. Then

$$ \begin{align*}\sum_{\ell \in L} \lvert\ell\rvert \leq \sum_{\ell \in L} \lvert\ell \cap P'\rvert + d(k-1)\lvert L\rvert + (k-1) m^{k-1} (\lvert P\rvert - \lvert P'\rvert ).\end{align*} $$

Proof. Let $P'' := P \backslash P'$ denote the set of elements $x \in P$ such that every $(k-1)$ -descendant of x has at most m children. Then we can rearrange the desired inequality as

$$ \begin{align*}\sum_{\ell \in L} \lvert \ell \cap P''\rvert \leq d(k-1)\lvert L\rvert + (k-1)m^{k-1} \lvert P''\rvert.\end{align*} $$

The quantity $\sum _{\ell \in L} \lvert \ell \cap P''\rvert $ is counting incidences $(x,\ell )$ where $\ell \in L$ and $x \in P'' \cap \ell $ .

Given $\ell \in L$ , call a point $x \in \ell $ low if it has no descending chain of length $k-1$ under it in $\ell $ . Every $\ell $ can contain at most $d(k-1)$ low points. Indeed, as $\ell $ is d-linear, it has at most d minimal elements. Removing them, we obtain a d-linear set $\ell _1 \subseteq \ell $ such that every point in it contains an element under it in $\ell $ , and $\ell _1$ itself has at most d minimal elements. Remove them to obtain a d-linear set $\ell _2 \subseteq \ell _1$ such that each point in it contains a descending chain of length $2$ under it in $\ell $ , and so on.

Hence each $\ell \in L$ contributes at most $d(k-1)$ incidences with its low points, giving a total contribution of at most $d(k-1)\lvert L\rvert $ to the sum. If x is not a low point on $\ell $ , then there are some $z_1 < \dotsb < z_{k-1} < x$ in $\ell $ , with each one a child of the next one. As L is a $K_{k,k}$ -free arrangement, among the sets $\ell \in L$ there are at most $k-1$ containing all these points. By the definition of $P''$ , for each $x \in P''$ there are at most $m^{k-1}$ choices for such tuples $(z_1, \dotsc , z_{k-1})$ . Hence x is incident to at most $(k-1)m^{k-1}$ sets $\ell \in L$ for which it is not low, and the total number of contributions of incidences in this case is at most $(k-1) m^{k-1} \lvert P''\rvert $ , so the claim follows.

Now we prove Theorem 4.3. Let t be a natural number to be chosen later and $m>0$ be another parameter to be chosen later. Define the subsets

$$ \begin{align*}P = P_0 \supset P_1 \supset \dots \supset P_t\end{align*} $$

of P by defining $P_0 := P$ , and for each $i=0,\dots ,t-1$ , defining $P_{i+1}$ to be the set of points in $P_i$ that have a $(k-1)$ -descendant with more than m children in $(P_i, <)$ . By Lemma 4.4, we have

$$ \begin{align*}\sum_{\ell \in L} \lvert\ell \cap P_i\rvert \leq \sum_{\ell \in L} \lvert\ell \cap P_{i+1}\rvert + d(k-1)\lvert L\rvert + (k-1)m^{k-1} (\lvert P_i\rvert - \lvert P_{i+1}\rvert )\end{align*} $$

for all $i=0,\dots ,t-1$ , and hence on telescoping,

$$ \begin{align*}\sum_{\ell \in L} \lvert\ell\rvert \leq \sum_{\ell \in L} \lvert\ell \cap P_t\rvert + d(k-1) t \lvert L\rvert + (k-1) m^{k-1} n_1.\end{align*} $$

Claim 4.5. Let x be a point in $P_t$ . Then it has at least $ \frac {m^t}{\left (k d^k\right )^{t-1}}$ distinct descendants in P.

Proof. By definition of $P_t$ there is some $(k-1)$ -descendant $x' \in P_{t-1}$ of x which has at least $ m$ children in $P_{t-1}$ . Let $S_{t-1} \subseteq P_{t-1}$ denote the set of children of $x'$ , so $\lvert S_{t-1}\rvert \geq m$ . By reverse induction for $i= t-1, t-2, \dotsc , 0$ , we choose sets $S_{i} \subseteq P_i$ of descendants of x so that $\lvert S_{i-1}\rvert \geq \frac {\lvert S_i\rvert m }{k d^k}$ . Then $\lvert S_0\rvert \geq \frac { m^t}{\left (k d^k\right )^{t-1}} $ , as wanted.

Let