1. Introduction
We fix $r \in \mathbb {N}_{\geq 2}$ and let $H = \left (V_1, \dotsc , V_r; E \right )$ be an rpartite and runiform hypergraph (or just an rhypergraph 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 rhypergraph $K_{k, \dotsc , k}$ 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 $\lvert E\rvert = O_{r,k} \left (n^{r  \frac {1}{k^{r1}}} \right )$ . 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 ominimal 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 longstanding unitdistance conjecture of Erdős (an almostlinear 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 axisparallel 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 almostlinear 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. 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. 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. 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. 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. More generally, given finitely many halfspaces $H_1, \dotsc , H_s$ 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
where $p \leq s \in \mathbb {N}$ and each $f_i: V^d \to V$ is a linear function – that is, of the form
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 rhypergraph 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 rhypergraph $\left (V_1, \dotsc , V_r; X \cap \prod _{i \in [r]} V_i \right )$ .
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 rhypergraph of description complexity $(s,t)$ and H is $K_{k, \dotsc , k}$ free, then
In particular, $\lvert E\rvert = O_{r,s,t,k,\varepsilon } \left ( n^{r1 + \varepsilon } \right )$ 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 $(d1)$ dimensional simplices in $\mathbb {R}^d$ .
One can get rid of the logarithmic factor entirely by restricting to the family of all finite rhypergraphs induced by a given $K_{k, \dotsc , k}$ free semilinear relation (as opposed to all $K_{k, \dotsc , k}$ free rhypergraphs 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 rhypergraph 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 $\lvert E\rvert = O_X\left (n^{r1}\right )$ .
This is Corollary 5.12 and follows from a more general Theorem 5.6 connecting linear Zarankiewicz bounds to a modeltheoretic notion of linearity of a firstorder 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 firstorder 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 pointline 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 ominimal 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 $\lvert X \cap \prod _{i \in [r]} V_i\rvert \neq O\left (n^{r1}\right )$ . 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 coordinatewise 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 firstorder 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 ominimal case (Corollary 5.11).
2. Upper bounds
2.1. Coordinatewise monotone functions and basic sets
For an integer $r\in \mathbb N_{>0}$ , by an rgrid (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 rgrid, S an arbitrary set and $f: B \to S$ a function. For $i \in [r]$ , set
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 rgrid and $(S,<)$ a linearly ordered set. A function $f\colon B\to S$ is coordinatewise monotone if for any $i\in [r]$ , $a,a'\in B^i$ and $b,b'\in B_i$ , we have
Remark 2.2. Let $B =B_1{{\times }\dotsb {\times }} B_r$ be an rgrid and $\Gamma $ an ordered abelian group. We say that a function $f\colon B\to \Gamma $ is quasilinear if there exist some functions $f_i\colon B_i\to \Gamma $ , $i\in [r]$ , such that
Then every quasilinear function is coordinatewise 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 quasilinear, and hence coordinatewise monotone.
Remark 2.4. Let B be a grid and $C \subseteq B$ a subgrid. If $f\colon B\to S$ is a coordinatewise monotone function, then the restriction $f{\restriction C}$ is a coordinatewise monotone function on C.
Definition 2.5. Let B be an rgrid. A subset $X\subseteq B$ is a basic set if there exists a linearly ordered set $(S,<)$ , a coordinatewise 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 coordinatewise 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 ‘coordinatesplitting’ presentation for basic sets:
Proposition 2.8. Let $B=B_1{{\times }\dotsb {\times }} B_r$ be an rgrid and $X\subseteq B$ a basic set. Then there is a linearly ordered set $(S,<)$ , a coordinatewise 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
Proof of Proposition 2.8. Assume that we are given a coordinatewise 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 $(b_1,\dotsc ,b_{r1}) <^r \left (b^{\prime }_1,\dotsc ,b^{\prime }_{r1}\right )$ if and only if
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
Claim 2.10. Set $a_1,a_2\in B^r$ and $b_1,b_2\in B_r$ .

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. 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 coordinatewise monotone.
2.2. Main theorem
Definition 2.12. Let $B=B_1{{\times }\dotsb {\times }} B_r$ be an rgrid.

1. Given $s \in \mathbb {N}$ , we say that a set $X\subseteq B$ has gridcomplexity s (in B) if X is the intersection of B with at most s basic subsets of B.
We say that X has finite gridcomplexity if it has gridcomplexity s for some $s \in \mathbb {N}$ .

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 gridcomplexity $0$ .
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
for some linear functions $f_i: V^d \to V, i \in [s]$ . Then each $f_i$ is coordinatewise monotone (Example 2.3), and hence each of the sets
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 gridcomplexity s.
Remark 2.14.

1. Let B be an rgrid and $A\subseteq B$ a subset of B of gridcomplexity s. If $C \subseteq B$ is a subgrid containing A, then A is also a subset of C of gridcomplexity s.

2. In particular, if $A\subseteq B$ is a subset of gridcomplexity s, then A is a subset of gridcomplexity $ 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 rgrid and set $n_i :=\lvert B_i\rvert $ . For $j\in \{0,\dotsc , r\}$ , we will denote by $\delta _j^r(B)$ the integer
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 rgrid B and $K_{k,\dotsc ,k}$ free subset $A \subseteq B$ of gridcomplexity s, we have
Moreover, we can take $\beta (r,s) := s\left (2^{r1}1\right )$ .
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 $\log ^\beta \left ( \delta ^r_{r1}(B)+1 \right )$ instead of $\log ^\beta \left ( \delta ^r_{r1}(B) \right )$ to include the case $\delta ^r_{r1}(B)\leq 1$ .
Remark 2.20. If, in Theorem 2.17, A is only assumed to be a union of at most t sets of gridcomplexity 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 rgrid 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 $\beta =s\left (2^{r1}1\right ) \in \mathbb N$ such that for any rgrid B and $K_{k,\dotsc ,k}$ free finite subset $A \subseteq B$ of gridcomplexity $\leq s$ , we have
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 gridcomplexity s of some rgrid B with $\delta _{r1}^r(B)\leq n$ .
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
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 gridcomplexity 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
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
For $l\in S$ , $i\in \{1,2\}$ and $\square \in \{ <,=,>, \leq , \geq \}$ , let
We choose $h\in S$ such that
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
Hence we conclude
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 rgrid.

1. For integers $t, u\in \mathbb N$ , we say that a subset $A\subseteq B$ is of split gridcomplexity $(t, u)$ if there are basic sets $X_1,\dotsc , X_{u} \subseteq B$ , a subset $A^r\subseteq B_1{{\times }\dotsb {\times }} B_{r1}$ of gridcomplexity 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. 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 rgrid B of split gridcomplexity $(t,u)$ with $\delta _{r1}^r(B)\leq n$ .
Remark 2.29.

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

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

3. If $A\subseteq B$ is of split gridcomplexity $(t, u)$ , then it is of gridcomplexity $t + u$ .
For the rest of the proof, we abuse notation slightly and refer to the split gridcomplexity of a set as simply the gridcomplexity. 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
We will use the following notations throughout the section:

• $B=B_1{{\times }\dotsb {\times }} B_r$ is a finite grid with $n=\delta ^r_{r1}(B)$ ;

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

• $B^r$ is the $(r1)$ grid $B^r :=B_1{{\times }\dotsb {\times }} B_{r1}$ ;

• $A^r \subseteq B^r$ is a subset of gridcomplexity 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 $\alpha =\alpha (r1, t,k)$ and $\beta =\beta (r1, t)$ such that $\lvert A^r\rvert \leq \alpha \delta ^{r1}_{r2}(B^r)\log ^\beta \left ( \delta ^{r1}_{r2}(B^r)+1\right )$ . In the second case, we have $\lvert A\rvert \leq \lvert B^r\rvert k=\delta ^{r1}_{r1}(B^r)k$ .
Since $n=\delta ^r_{r1}(B)=\delta ^{r1}_{r1}(B^r)+\delta ^{r1}_{r2}(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 $x^r := (x_1, \dotsc , x_{r1})$ . By Proposition 2.8, we can choose a finite linear order $(S,<)$ , a coordinatewise monotone function $f^r\colon B^r\to S$ and a function $f_r\colon B_r\to S$ so that
Moreover, by Remark 2.9 we may assume without loss of generality that the coordinatewise monotone function defining $X_u$ is given by
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
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
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. $A^r$ is an $f^r$ strip, and $A_r$ is an $f_r$ strip.

2. Every $f^r$ strip is a subset of the $(r1)$ grid $B^r$ of gridcomplexity $t+2$ (using Remark 2.7).

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. We say that a subset $H\subseteq B$ is an fgrid 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. If $H=H^r\times H_r$ is an fgrid, we set
$$ \begin{align*} \Delta(H) :=\lvert H^r\rvert + \delta^{r1}_{r2}(H^r)\lvert H_r\rvert \text{ (see Definition~2.21 for } \delta^{r1}_{r2}). \end{align*} $$Note that if H is a subgrid of B, then $\Delta (H)=\delta ^r_{r1}(H)$ . 
3. For an fgrid 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 fgrid H, if the set $A_H$ is $K_{k,\dotsc ,k}$ free then
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 fgrid $H := A^r\times A_r$ (so $A_H = A$ ) and get
It is easy to see that $\Delta (A^r\times A_r)\leq \delta ^r_{r1}(B)$ , 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 fgrids $H \subseteq B$ 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
Let $H=H^r\times H_r$ be an fgrid with $\Delta (H)\leq m$ .
For $l\in S$ and $\square \in \{ <,=,>, \leq , \geq \}$ , define
and
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 fgrid.
Claim 2.35. There is $h\in S$ such that
Proof. Set $\delta :=\delta ^{r1}_{r2}(H^r)$ .
Let h be the minimal element in $f^r(H^r)\cup f_r(H_r)$ with
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 ^{r1}_{r2}\left (H^{r,< h}\right ), \delta ^{r1}_{r2}\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:
It follows that
Hence, by the choice of h and using Remark 2.32(2),
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. $\beta (2,s) \leq s$ for all $s \in \mathbb {N}$ .

2. $\beta '(r,t, 0) \leq \beta (r1,t)$ for all $r \geq 3$ and $t \in \mathbb {N}$ .

3. $\beta '(r,t,u) \leq \beta '(r, t+2, u1) + 1$ 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), $\beta (r,s) \leq \beta (r1, 2s) + s$ for every $r \geq 3$ and $s \geq 1$ . Iterating this, we get $\beta (r,s) \leq \beta \left (2, 2^{r2}s\right ) + s \sum _{i=0}^{r3}2^i$ . Using (1), this implies $\beta (r,s) \leq s \sum _{i=0}^{r2}2^i = s\left (2^{r1}1\right )$ for all $r \geq 3, s \geq 1$ . Hence, by Remark 2.27 and (1) again, $\beta (r,s) \leq s\left (2^{r1}1\right )$ 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 $\beta (r,s) := s\left (2^{r1}1\right )$ satisfying the following: for any semilinear $K_{k, \dotsc , k}$ free rhypergraph $H = (V_1, \dotsc , V_r;E)$ of description complexity $(s,t)$ (see Definition 1.7), taking $V:= \prod _{i \in [r]}V_i$ we have
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 gridcomplexity $\leq s$ (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) halfspaces 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
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 halfspace.
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,
Then the incidence relation between points in $\mathbb {R}^d$ and polytopes in $\mathcal {F}$ can be identified with the semilinear set
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 $2d$ halfspaces 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 ordertype over $\bar {r}$ if
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 quantifierfree 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 ddimensional open boxes with axisparallel sides, with I incidences between P and B.

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. 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 ordertype 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 pointbox incidence configurations in a higherdimensional 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. there exists a set of points $P^{d1} \subseteq \mathbb {R}^{d1}$ with $\left \lvert P^{d1}\right \rvert = n_1$ and a set of $(d1)$ dimensional boxes $B^{d1}$ with $\left \lvert B^{d1}\right \rvert = n_2$ , with m incidences between them and the incidence graph $K_{2,2}$ free; and

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 ddimensional 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 ddimensional 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 $1 \leq j \leq d1$ , all points in $P^{d1}$ have pairwise distinct jth coordinates; and none of the points is on the border of any of the boxes. Write $P^{d1}$ as $p_1, \dotsc , p_{n_1}$ . Let $\bar {r}$ be the tuple listing all corners of all boxes in $B^{d1}$ .
Using this, for each $p_i$ we can choose a very small $(d1)$ 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 ordertype as $(p_1, \dotsc , p_{n_1})$ over $\bar {r}$ . In particular, every $\beta _i$ is pairwise disjoint, and the incidence graph between $P^{d1}$ and $B^{d1}$ is isomorphic to the incidence graph between $\left (p^{\prime }_i, \dotsc , p^{\prime }_{n_1}\right )$ and $B^{d1}$ 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 $B^{d1}$ 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 $c_l \in B^{d1}$ , we consider the ddimensional 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 $1 \leq j \leq n^{\prime }_1$ , $(\beta _i \times \mathbb {R}) \cap \left (\mathbb {R}^{d1} \times I_j\right )$ 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 $d1$ coordinates is in $\beta _i$ . Hence the incidence graph between P and $B_j$ is isomorphic to the incidence graph between $P^{d1}$ and $B^{d1}$ 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 $\left (P^{d1}, B^{d1}\right )$ , 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 $B^{d1}$ 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, $n_1^j n^{\prime }_2 + j n_1^{j1} n^{\prime }_1 n_2$ dyadic boxes (Remark 3.4) and $n_1^j m' + j n_1^{j1} n^{\prime }_1 m$ 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 $d1$ 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 ‘pointbox’ 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 axisparallel 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 dlinear 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 dlinear if it contains no antichains of size greater than d, and $(P, \leq )$ is locally dlinear if any interval $[a,b] = \{ x \in P : a \leq x \leq b \}$ is dlinear.
Note that dlinearity 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 $k1$ 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 dlinear and L is a $K_{k,k}$ free arrangement of dlinear subsets of P. Then
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 tdescendant 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 tdescendant of x if it is a strict sdescendant 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 dlinear subsets of P and let $m> 0$ . Let $P'$ denote the set of all elements in P which have a $(k1)$ descendant with more than m children. Then
Proof. Let $P'' := P \backslash P'$ denote the set of elements $x \in P$ such that every $(k1)$ descendant of x has at most m children. Then we can rearrange the desired inequality as
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 $k1$ under it in $\ell $ . Every $\ell $ can contain at most $d(k1)$ low points. Indeed, as $\ell $ is dlinear, it has at most d minimal elements. Removing them, we obtain a dlinear 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 dlinear 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 $d(k1)$ incidences with its low points, giving a total contribution of at most $d(k1)\lvert L\rvert $ to the sum. If x is not a low point on $\ell $ , then there are some $z_1 < \dotsb < z_{k1} < x$ 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 $k1$ containing all these points. By the definition of $P''$ , for each $x \in P''$ there are at most $m^{k1}$ choices for such tuples $(z_1, \dotsc , z_{k1})$ . Hence x is incident to at most $(k1)m^{k1}$ sets $\ell \in L$ for which it is not low, and the total number of contributions of incidences in this case is at most $(k1) m^{k1} \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
of P by defining $P_0 := P$ , and for each $i=0,\dots ,t1$ , defining $P_{i+1}$ to be the set of points in $P_i$ that have a $(k1)$ descendant with more than m children in $(P_i, <)$ . By Lemma 4.4, we have
for all $i=0,\dots ,t1$ , and hence on telescoping,
Claim 4.5. Let x be a point in $P_t$ . Then it has at least $ \frac {m^t}{\left (k d^k\right )^{t1}}$ distinct descendants in P.
Proof. By definition of $P_t$ there is some $(k1)$ descendant $x' \in P_{t1}$ of x which has at least $ m$ children in $P_{t1}$ . Let $S_{t1} \subseteq P_{t1}$ denote the set of children of $x'$ , so $\lvert S_{t1}\rvert \geq m$ . By reverse induction for $i= t1, t2, \dotsc , 0$ , we choose sets $S_{i} \subseteq P_i$ of descendants of x so that $\lvert S_{i1}\rvert \geq \frac {\lvert S_i\rvert m }{k d^k}$ . Then $\lvert S_0\rvert \geq \frac { m^t}{\left (k d^k\right )^{t1}} $ , as wanted.
Let