# Vertex Form By Partial Factoring Attending Vertex Form By Partial Factoring Can Be A Disaster If You Forget These 1 Rules

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 aloof an r-hypergraph for brevity) with acme sets $V_1, dotsc , V_r$ accepting $lvert V_irvert = n_i$ , (hyper-) bend set E and a absolute cardinal $n = sum _{i=1}^r n_i$ of vertices.

Zarankiewicz’s botheration asks for the best cardinal of edges in such a hypergraph H (as a action of $n_1, dotsc , n_r$ ) bold that it does not accommodate the complete r-hypergraph $K_{k, dotsc , k}$ with $k> 0$ a anchored cardinal of vertices in anniversary part. The afterward classical high apprenticed is due to Kővári, Sós and Turán [Reference Kővári, Sós and Turán14] for $r=2$ and Erdős [Reference Erdős9] for a accepted r: if H is $K_{k, dotsc , k}$ -free, afresh $lvert Ervert = O_{r,k} left (n^{r – frac {1}{k^{r-1}}} right )$ . A probabilistic architecture in [Reference Erdős9] additionally shows that the backer cannot be essentially improved.

However, stronger apprenticed are accepted for belted families of hypergraphs arising in geometric settings. For example, if H is the accident blueprint of a set of $n_1$ credibility and $n_2$ curve in $mathbb {R}^2$ , afresh H is $K_{2,2}$ -free, and the Kővári–Sós–Turán Acceptance implies $lvert Ervert = Oleft ( n^{3/2}right )$ . The Szemerédi–Trotter Acceptance [Reference Szemerédi and Trotter20] improves this and gives the optimal apprenticed $lvert Ervert = Oleft (n^{4/3}right )$ . Added generally, [Reference Fox, Pach, Sheffer, Suk and Zahl12] gives bigger apprenticed for semialgebraic graphs of belted description complexity. This is generalised to semialgebraic hypergraphs in [Reference Do8]. In a altered direction, the after-effects in [Reference Fox, Pach, Sheffer, Suk and Zahl12] are generalised to graphs apprenticed in o-minimal structures in [Reference Basu and Raz2] and, added generally, in distal structures in [Reference Chernikov, Galvin and Starchenko4].

A accompanying awful nontrivial botheration is to accept back the apprenticed offered by the after-effects in the above-mentioned branch are sharp. Back H is the accident blueprint of $n_1$ credibility and $n_2$ circles of assemblage ambit in $mathbb {R}^2$ , the best accepted high apprenticed is $lvert Ervert =Oleft (n^{4/3}right )$ , accurate in [Reference Spencer, Szemerédi and Trotter19] and additionally adumbrated by the accepted apprenticed for semialgebraic graphs. Any advance to this apprenticed will be a footfall against absolute the abiding unit-distance acceptance of Erdős (an almost-linear apprenticed of the anatomy $lvert Ervert =Oleft (n^{1 c/log log n}right )$ will absolutely boldness it).

This cardboard was originally motivated by the afterward accident problem: Let H be the accident blueprint of a set of $n_1$ credibility and a set of $n_2$ solid rectangles with axis-parallel abandon (which we accredit to as boxes) in $mathbb {R}^2$ . Bold that H is $K_{2,2}$ -free – that is, no two credibility accord to two rectangles accompanying – what is the best cardinal of incidences $lvert Ervert $ ? In the afterward theorem, we access an almost-linear apprenticed (which is abundant stronger than the apprenticed adumbrated by the above accepted aftereffect for semialgebraic graphs) and authenticate that it is abutting to optimal:

Problem 1.1. While the apprenticed for dyadic boxes is tight, we leave it as an accessible botheration to abutting the gap amid the high and lower apprenticed for approximate boxes.

Theorem (A.1) admits the afterward generalisation to college ambit and added accepted polytopes:

Theorem (B).

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

2. Added generally, accustomed finitely abounding half-spaces $H_1, dotsc , H_s$ in $mathbb {R}^d$ , let $mathcal {F}$ be the ancestors of all accessible polytopes in $mathbb {R}^d$ cut out by approximate translates of $H_1, dotsc , H_s$ . Afresh for any set P of $n_1$ credibility in $mathbb {R}^d$ and any set F of $n_2$ polytopes in $mathcal {F}$ , if the accident blueprint on $P times F$ is $K_{k,k}$ -free, afresh it contains at best $O_{k,s}left ( n log ^{s} n right )$ incidences (Corollary 2.37).

Problem 1.3. What is the optimal apprenticed on the ability of $log n$ in Acceptance (B)? In particular, does it absolutely accept to abound with the ambit d?

The high apprenticed in Theorems (A.1) and (B) are acquired as actual applications of a accepted high apprenticed for Zarankiewicz’s botheration for semilinear hypergraphs of belted description complexity.

We focus on the case $V=R = mathbb {R}$ in the introduction, back these are absolutely the semialgebraic sets that can be authentic application alone beeline polynomials.

We accent that there is no brake on the ambit $d_i$ in this definition. We access the afterward accepted high bound:

In particular, $lvert Ervert = O_{r,s,t,k,varepsilon } left ( n^{r-1 varepsilon } right )$ for any $varepsilon>0$ in this case. For a added absolute statement, see Aftereffect 2.36 (in particular, the assurance of the connected in $O_{r,s,t,k}$ on k is at best linear).

One can get rid of the logarithmic agency absolutely by akin to the ancestors of all apprenticed r-hypergraphs induced by a accustomed $K_{k, dotsc , k}$ -free semilinear affiliation (as against to all $K_{k, dotsc , k}$ -free r-hypergraphs induced by a accustomed approximate semilinear relation, as in Acceptance (C)).

This is Aftereffect 5.12 and follows from a added accepted Acceptance 5.6 abutting beeline Zarankiewicz apprenticed to a model-theoretic angle of breadth of a first-order anatomy (in the faculty that the matroid accustomed by the algebraic cease abettor behaves like the beeline amount in a agent space, as against to the algebraic cease in an algebraically bankrupt acreage – see Analogue 5.3).

In particular, for every $K_{k,k}$ -free semilinear affiliation $X subseteq mathbb {R}^{d_1} times mathbb {R}^{d_2}$ (equivalently, X apprenticed with ambit in the first-order anatomy $(mathbb {R}, <, )$ by Remark 1.6) we accept $lvert X cap (V_1 times V_2)rvert = O(n)$ for all $V_i subseteq mathbb {R}^{d_i}_i$ , $lvert V_irvert = n_i$ , $n = n_1 n_2$ . One the added hand, by optimality of the Szemerédi–Trotter bound, for the semialgebraic $K_{2,2}$ -free point-line accident blueprint $X = left {(x_1,x_2; y_1, y_2) in mathbb {R}^4 : x_2 = y_1 x_1 y_2 right }subseteq mathbb {R}^2 times mathbb {R}^2$ we accept $lvert X cap (V_1 times V_2)rvert = Omega left (n^{frac {4}{3}}right )$ . Note that in adjustment to ascertain it we use both accession and multiplication – that is, the acreage structure. This is not coincidental; as a aftereffect of the trichotomy acceptance in o-minimal structures [Reference Peterzil and Starchenko18], we beam that the abortion of a beeline Zarankiewicz apprenticed consistently allows us to balance the acreage in a apprenticed way (Corollary 5.11). In the semialgebraic case, we accept the afterward aftereffect that is accessible to accompaniment (Corollary 5.14):

We achieve with a abrupt overview of the paper.

In Area 2 we acquaint a added accepted chic of hypergraphs apprenticed in agreement of coordinate-wise banausic functions (Definition 2.1) and prove an high Zarankiewicz apprenticed for it (Theorem 2.17). Theorems (A.1), (B) and (C) are afresh deduced from it in Area 2.5.

In Area 3 we prove Acceptance (A.3) by establishing a lower apprenticed on the cardinal of incidences amid credibility and dyadic boxes on the plane, demonstrating that the logarithmic agency is assertive (Proposition 3.5).

In Area 4, we authorize Acceptance (A.2) by accepting a stronger apprenticed on the cardinal of incidences with dyadic boxes on the even (Theorem 4.7). We use a altered argument, relying on a assertive fractional adjustment specific to the dyadic case, to abate from $log ^4(n)$ accustomed by the accepted acceptance to $log (n)$ . Up to a connected factor, this implies the aforementioned apprenticed for incidences with accepted boxes back one counts alone incidences that are belted abroad from the apprenticed (Remark 4.8).

Finally, in Area 5 we prove a accepted Zarankiewicz apprenticed for apprenticed relations in abominably locally modular geometric first-order structures (Theorem 5.6), deduce Acceptance (D) from it (Corollary 5.12) and beam how to balance a absolute bankrupt acreage from the abortion of Acceptance (D) in the o-minimal case (Corollary 5.11).

For an accumulation $rin mathbb N_{>0}$ , by an r-grid (or a grid, if r is bright from the context) we beggarly a cartesian artefact $B=B_1{{times }dotsb {times }} B_r$ 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, afresh by a subgrid we beggarly a subset $C subseteq B$ of the anatomy $C=C_1{{times }dotsb {times }} C_r$ for some $C_i subseteq B_i$ .

Let B be an r-grid, S an approximate 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 bump maps.

For $a in B^i $ and $b in B_i$ , we address $a oplus _i b$ for the aspect $c in B$ with $pi ^i(c) = a$ and $pi _i(c) = b$ . In particular, back $i = r$ , $a oplus _r b = (a, b)$ .

We accept the afterward ‘coordinate-splitting’ presentation for basal sets:

Proof of Antecedent 2.8. Accept that we are accustomed a coordinate-wise banausic action $fcolon Bto S$ and $lin S$ with $X= left { bin B colon f(b) < lright }$ .

For $iin [r]$ , let $leq _i$ be the preorder on $B_i$ induced by f – namely, for $b,b’in B_i$ we set $bleq _i b’$ if and alone if for some (equivalently, any) $ain B^i$ we accept $f(a oplus _i b)leq f(a oplus _i b’)$ .

Quotienting $B_i$ by the adequation affiliation agnate to the preorder $leq _i$ if needed, we may accept that anniversary $leq _i$ is absolutely a beeline order.

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

Define $T := B^r dot cup B_r$ , area $dot cup $ denotes the break union. Acutely $<^r$ is a austere fractional adjustment on T – that is, a transitive and antisymmetric (hence irreflexive) relation.

For any $b^rin B^r$ and $b_rin B_r$ , we define

Let $triangleleft ^t$ be the transitive cease of $triangleleft $ . It follows from the above-mentioned affirmation that $triangleleft ^t=triangleleft cup triangleleft {circ }triangleleft $ . Added explicitly, for $b_1,b_2 in B_r$ , we accept $b_1 triangleleft ^t b_2$ if $b_2 <_r b_1$ , and for $a_1,a_2 in B^r$ , we accept $a_1 triangleleft ^t a_2$ if $f(a_1 oplus b) < l < f(a_2 oplus b)$ for some $b in B_r$ . It is not adamantine to see afresh that $triangleleft ^t$ is antisymmetric, and appropriately it is a austere fractional adjustment on T.

Claim 2.11. The abutment $<^r cup triangleleft ^t$ is a austere fractional adjustment on T.

Proof. We aboriginal appearance transitivity. Note that $<^r$ and $triangleleft ^t$ are both transitive, so it suffices to appearance for $x, y, z in T$ that if either $x <^r y triangleleft ^t z$ or $x triangleleft ^t y <^r z$ , afresh $x triangleleft ^t z$ . Furthermore, back $triangleleft ^t=triangleleft cup triangleleft {circ }triangleleft $ , we may bind our absorption to the afterward cases: If $a_1 <^r a_2triangleleft b$ with $a_1,a_2in B^r$ and $bin B_r$ , afresh $f(a_1 oplus _r b)<f(a_2 oplus _r b)<l$ , and so $a_1triangleleft b$ . If $btriangleleft a_1 <^r a_2$ with $a_1,a_2in B^r$ and $bin B_r$ , afresh $f(a_2 oplus _r b)>f(a_1 oplus _r b)geq l$ , and so $btriangleleft a_2$ .

To analysis antisymmetry, accept $a_1 <^r a_2$ and $a_2 triangleleft ^t a_1$ . Back $a_1,a_2in B^r$ , we accept $a_2triangleleft b triangleleft a_1$ for some $bin B_r$ . We accept $f(a_1 oplus _r b)geq l> f(a_2 oplus _r b)$ , contradicting $a_1<^r a_2$ .

Finally, let $prec $ be an approximate beeline adjustment on $T=B^rdot cup B_r$ extending $<^r cup triangleleft ^t$ . Back $prec $ extends $triangleleft $ , for $ain B^r$ and $bin B_r$ we accept $(a,b)in X$ if and alone if $aprec b$ .

We booty $f^rcolon B^rto T$ and $f_rcolon B_rto T$ to be the character maps. Back $prec $ extends $<^r$ , the map $f^r$ is coordinate-wise monotone.

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

We can now accompaniment the capital theorem:

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

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

If B is a apprenticed r-grid and $Asubseteq B$ , afresh acutely $delta ^r_j(A)leq delta ^r_j(B)$ . Appropriately Acceptance 2.17 is agnate to the following:

Then Antecedent 2.22 can be restated as follows:

In the blow of the area we prove Antecedent 2.24 by consecration on r, area for anniversary r it is accepted by consecration on s. We will use the afterward simple ceremony bound:

Let $B=B_1{times } B_2$ be a apprenticed filigree and $Asubseteq B$ a subset of grid-complexity s. We will advance by consecration on s.

If $s=0$ , afresh $A=B_1times B_2$ . If A is $K_{k,k}$ -free, afresh one of the sets $B_1, B_2$ charge accept admeasurement at best k. Appropriately $lvert Arvert leq k(lvert B_1rvert lvert B_2rvert )=kdelta ^2_1(B)$ .

Thus

Assume now that the acceptance is accepted for $r=2$ and all $s'<s$ . Ascertain $n_1 :=lvert B_1rvert $ , $n_2 :=lvert B_2rvert $ and $n :=delta ^2_1(B)=n_1 n_2$ .

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

By Antecedent 2.8, we can accept a apprenticed beeline adjustment $(S,<)$ and functions $f_1colon B_1to S$ and $f_2colon B_2to S$ so that

For $lin S$ , $iin {1,2}$ and $square in { <,=,>, leq , geq }$ , let

We accept $hin S$ such that

For example, we can booty h to be the basal aspect 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 consecration antecedent on s and application Actuality 2.26 and Remark 2.25, we access $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 abject case $r=2$ .

We fix $r in mathbb {N}_{geq 3}$ and accept that Antecedent 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 apprenticed r-grid.

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

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

For the blow of the proof, we corruption characters hardly and accredit to the breach grid-complexity of a set as artlessly the grid-complexity. To complete the consecration footfall we will prove the afterward proposition:

Proposition 2.30. For any integers $t,ugeq 0, kgeq 2, r geq 3$ , 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 afterward notations throughout the section:

We advance by consecration on u.

The abject case $u=0$ of Antecedent 2.30.

In this case, $A=A^rtimes A_r$ . If A is $K_{k,dotsc ,k}$ -free, afresh either $A^r$ is $K_{k,dotsc ,k}$ -free or $lvert A_rrvert <k$ .

In the aboriginal case, by the consecration antecedent on r, there are $alpha =alpha (r-1, t,k)$ and $beta =beta (r-1, t)$ such that $lvert A^rrvert leq alpha delta ^{r-1}_{r-2}(B^r)log ^beta left ( delta ^{r-1}_{r-2}(B^r) 1right )$ . In the additional case, we accept $lvert Arvert leq lvert B^rrvert k=delta ^{r-1}_{r-1}(B^r)k$ .

Since $n=delta ^r_{r-1}(B)=delta ^{r-1}_{r-1}(B^r) delta ^{r-1}_{r-2}(B^r) lvert B_rrvert $ , the cessation of the antecedent follows with $alpha ‘ := alpha , beta ‘ := beta $ .

Induction footfall of Antecedent 2.30.

We accept now that the antecedent 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 Antecedent 2.8, we can accept a apprenticed beeline adjustment $(S,<)$ , a coordinate-wise banausic action $f^rcolon B^rto S$ and a action $f_rcolon B_rto S$ so that

Moreover, by Remark 2.9 we may accept after accident of generality that the coordinate-wise banausic action defining $X_u$ is accustomed by

Definition 2.31. Accustomed an approximate 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 $l_1,l_2in S$ , $triangleleft _1, triangleleft _2in { <,leq }$ . Likewise, accustomed an approximate 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 $l_1,l_2in S$ , $triangleleft _1, triangleleft _2in { <,leq }$ . If $C^r = A^r$ or $C_r = A_r$ , we artlessly say an $f^r$ -strip or $f_r$ -strip, respectively.

The consecration footfall for Antecedent 2.30 will chase from the afterward proposition:

Proposition 2.34. For all integers $kgeq 2, r geq 3$ , 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

We should accent 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 anchored t and u.

Given Antecedent 2.34, we can administer it to the f-grid $H := A^rtimes A_r$ (so $A_H = A$ ) and get

It is accessible to see that $Delta (A^rtimes A_r)leq delta ^r_{r-1}(B)$ , and appropriately Antecedent 2.30 follows with the aforementioned $alpha ‘$ and $beta ‘$ .

We advance with the affidavit of Antecedent 2.34:

Proof of Antecedent 2.34. Fix $min mathbb N$ , and let $L(m)$ be the acute admeasurement of a $K_{k,dotsc ,k}$ -free set $A_H$ amid all f-grids $H subseteq B$ with $Delta (H)leq m$ . We charge to appearance that for some $alpha ‘=alpha ‘(k)in mathbb {R}$ and $beta ‘ in mathbb N$ we have

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

For $lin S$ 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 artefact is an f-grid.

Claim 2.35. There is $hin S$ such that

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

It follows that

Hence, by the best of h and application Remark 2.32(2),

Applying the consecration antecedent on u and application Actuality 2.26, we access $L(m)leq alpha ‘ m log ^{beta ‘}(m 1)$ for some $alpha ‘=alpha ‘(k)in mathbb {R}$ and $beta ‘in mathbb N$ .

This finishes the affidavit of Antecedent 2.34, and appropriately of the consecration footfall of Antecedent 2.24.

Finally, analytical the proof, we accept apparent the following:

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

2. $beta ‘(r,t, 0) leq beta (r-1,t)$ for all $r geq 3$ and $t in mathbb {N}$ .

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 accept $beta (r,s) leq beta ‘(r,0,s) leq beta ‘(r, 2s, 0) s$ . 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 $beta (r,s) leq beta left (2, 2^{r-2}sright ) s sum _{i=0}^{r-3}2^i$ . Application (1), this implies $beta (r,s) leq s sum _{i=0}^{r-2}2^i = sleft (2^{r-1}-1right )$ for all $r geq 3, s geq 1$ . Hence, by Remark 2.27 and (1) again, $beta (r,s) leq sleft (2^{r-1}-1right )$ for all $r geq 2, s geq 0$ .

We beam several actual applications of Acceptance 2.17, starting with the afterward apprenticed for semilinear hypergraphs:

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

Proof. By assumption, the bend affiliation E can be authentic by a abutment of t sets, anniversary of which is authentic by s beeline equalities and inequalities, appropriately of grid-complexity $leq s$ (see Archetype 2.13). The cessation follows by Acceptance 2.17 and Remark 2.20.

As a appropriate case with $r=2$ , this implies a apprenticed for the afterward accident problem:

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 accessible or a bankrupt half-space.

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

Then the accident affiliation amid credibility in $mathbb {R}^d$ and polytopes in $mathcal {F}$ can be articular with the semilinear set

defined by s beeline inequalities. The cessation now follows by Aftereffect 2.36 with $r=2$ .

In particular, we get a apprenticed for the aboriginal catechism that motivated this paper.

While we do not apperceive if the apprenticed $beta (2,s) leq s$ in Acceptance 2.17 is optimal, in this area we appearance that at atomic the logarithmic agency is assertive already for the accident affiliation amid credibility and dyadic boxes in $mathbb {R}^2$ .

We alarm a hardly added accepted architecture first. Fix $d in mathbb {N}_{>0}$ .

Definition 3.1. Accustomed apprenticed 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 $bar {q}$ accept the aforementioned order-type 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 added words, the tuples $left (p_{i,j} : 1 leq i leq n, 1leq j leq dright )$ and $left (q_{i,j} : 1 leq i leq n, 1leq j leq dright )$ accept the aforementioned quantifier-free blazon over the set $left {r_{i,j} : 1 leq i leq m, 1 leq j leq d right }$ in the anatomy $(mathbb {R}, <)$ .

We accept the afterward antecedent for accumulation point-box accident configurations in a higher-dimensional space:

Lemma 3.3. Accustomed any $d,n_1,n_2,n^{prime }_1, n^{prime }_2, m,m’ in mathbb {N}_{>0}$ , accept that:

1. there exists a set of credibility $P^{d-1} subseteq mathbb {R}^{d-1}$ 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 amid them and the accident blueprint $K_{2,2}$ -free; and

2. there exists a set of credibility $P^d subseteq mathbb {R}^d$ with $left lvert P^dright rvert = n^{prime }_1$ and a set of d-dimensional boxes $B^d$ with $left lvert B^dright rvert =n^{prime }_2$ , with $m’$ incidences amid them and the accident blueprint $K_{2,2}$ -free.

Then there exists a set of credibility $P subseteq mathbb {R}^d$ with $lvert Prvert = n_1 n^{prime }_1$ and a set of d-dimensional boxes B with $lvert Brvert = n_1n^{prime }_2 n^{prime }_1n_2$ , so that there are $n_1m’ m n^{prime }_1$ incidences amid P and B and their accident blueprint is still $K_{2,2}$ -free.

Proof. By Remark 3.2(1) we may accept that for every $1 leq j leq d$ , all credibility in $P^d$ accept pairwise audible jth coordinates; for every $1 leq j leq d-1$ , all credibility in $P^{d-1}$ accept pairwise audible jth coordinates; and none of the credibility is on the apprenticed of any of the boxes. Address $P^{d-1}$ as $p_1, dotsc , p_{n_1}$ . Let $bar {r}$ be the tuple advertisement all corners of all boxes in $B^{d-1}$ .

Using this, for anniversary $p_i$ we can accept a actual baby $(d-1)$ -dimensional box $beta _i$ with $p_i in beta _{i}$ and such that for any best of credibility $p^{prime }_i in beta _i, 1 leq i leq n_1$ , we accept that $left (p^{prime }_1, dotsc , p^{prime }_{n_1}right )$ has the aforementioned order-type as $(p_1, dotsc , p_{n_1})$ over $bar {r}$ . In particular, every $beta _i$ is pairwise disjoint, and the accident blueprint amid $P^{d-1}$ and $B^{d-1}$ is isomorphic to the accident blueprint amid $left (p^{prime }_i, dotsc , p^{prime }_{n_1}right )$ and $B^{d-1}$ by Remark 3.2(2).

Contracting and advice while befitting the dth alike unchanged, for anniversary $1 leq i leq n_1$ we can acquisition a archetype $left (P^{d}_i, B^d_iright )$ of the agreement $left (P^d, B^dright )$ absolutely independent in the box $beta _i times mathbb {R}$ – that is,

Set $P := bigcup _{1 leq i leq n_1} P^d_i$ and $B’ := bigcup _{1 leq i leq n_1} B^d_i$ ; afresh $lvert Prvert = n_1n^{prime }_1, lvert B’rvert = n_1n^{prime }_2$ and there are $n_1m’$ incidences amid 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 credibility in $P^d$ are pairwise disjoint, for anniversary $1 leq j leq n^{prime }_1$ we can accept a baby breach $I_j subseteq mathbb {R}$ 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 anniversary $1 leq j leq n^{prime }_1$ and $c_l in B^{d-1}$ , we accede the d-dimensional box $c_{j,l} :=c_l times I_j$ . Ascertain $B_j := left {c_{j,l} : 1 leq l leq n_2 right }$ . For anniversary $1 leq i leq n_1$ and $1 leq j leq n^{prime }_1$ , $(beta _i times mathbb {R}) cap left (mathbb {R}^{d-1} times I_jright )$ contains absolutely one point $q_{i,j}$ (given by the archetype of $q_{j}$ in $P_i^d$ ), and the bump $q^{prime }_{i,j}$ of $q_{i,j}$ assimilate the aboriginal $d-1$ coordinates is in $beta _i$ . Appropriately the accident blueprint amid P and $B_j$ is isomorphic to the accident blueprint amid $P^{d-1}$ and $B^{d-1}$ by the best of the $beta _i$ s, and in accurate the cardinal of incidences is m.

Finally, ascertain $B := B’ cup bigcup _{1 leq j leq n^{prime }_1} B_j$ ; afresh $lvert Brvert = n_1n^{prime }_2 n^{prime }_1n_2$ . 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 $jneq j’$ . It is now not adamantine to analysis that the accident blueprint amid P and B is $K_{2,2}$ -free, by architecture and the assumptions of $K_{2,2}$ -freeness of $left (P^d,B^dright )$ and $left (P^{d-1}, B^{d-1}right )$ , and that there are $n_1m’ mn^{prime }_1$ incidences amid P and B.

Proof. Accustomed d, accept that there abide $K_{2,2}$ -free ‘point–dyadic box’ configurations acceptable Antecedent 3.3(1) and (2) for some ambit $d, n_1, n_2, n^{prime }_1, n^{prime }_2, m, m’$ . Afresh for any $j in mathbb {N}$ , we can iterate the antecedent j times and acquisition a $K_{2,2}$ -free ‘point–dyadic box’ agreement 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 alpha with $n_1 = ell , n_2 = 1, m=ell $ (one dyadic breach absolute $n_1$ credibility in $mathbb {R}$ ) and $n^{prime }_1=1, n^{prime }_2=0, m’ = 0$ (one point and aught dyadic boxes in $mathbb {R}^2$ ). Demography $j := ell $ , we afresh acquisition a $K_{2,2}$ -free agreement with $ell ^{ell }$ points, $ell ^ell $ dyadic boxes and $ell ell ^{ell }$ incidences. Appropriately for $n := k^k$ , we accept n points, n boxes and $Omega left (n frac {log n}{log log n} right )$ incidences.

In this area we strengthen the apprenticed on the cardinal of incidences with rectangles on the even with axis-parallel abandon accustomed by Aftereffect 2.38 – that is, $O_{k} left ( n log ^{4} n right )$ – in the appropriate case of dyadic rectangles, application a altered altercation (which relies on a assertive fractional adjustment specific to the dyadic case).

Throughout this section, let $(P, leq )$ be a partially ordered set of admeasurement at best $n_1$ , and let L be a accumulating of subsets of P (possibly with repetitions) of admeasurement at best $n_2$ . As before, we let $n = n_1 n_2$ .

Note that d-linearity is preserved beneath removing credibility from P.

Observe that if one removes any cardinal of credibility from P or removes any cardinal of sets from L, one still obtains a $K_{k,k}$ -free arrangement. We now accompaniment the capital acceptance of this section:

To prove this theorem, we aboriginal charge some definitions and a lemma. If $x in P$ , ascertain a ancestor of x to be an aspect $y in P$ with $y>x$ and no aspect amid x and y, and analogously ascertain a adolescent of x to be an aspect $z in P$ with $z<x$ and no aspect amid z and x. We say that z is a austere 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 adolescent of $z_i$ , and that z is a t-descendant of x if it is a austere s-descendant for some $0 leq s leq t$ .

Proof. Let $P” := P backslash P’$ denote the set of elements $x in P$ such that every $(k-1)$ -descendant of x has at best m children. Afresh we can adapt the adapted asperity as

The abundance $sum _{ell in L} lvert ell cap P”rvert $ is counting incidences $(x,ell )$ area $ell in L$ and $x in P” cap ell $ .

Given $ell in L$ , alarm a point $x in ell $ low if it has no bottomward alternation of breadth $k-1$ beneath it in $ell $ . Every $ell $ can accommodate at best $d(k-1)$ low points. Indeed, as $ell $ is d-linear, it has at best d basal elements. Removing them, we access a d-linear set $ell _1 subseteq ell $ such that every point in it contains an aspect beneath it in $ell $ , and $ell _1$ itself has at best d basal elements. Remove them to access a d-linear set $ell _2 subseteq ell _1$ such that anniversary point in it contains a bottomward alternation of breadth $2$ beneath it in $ell $ , and so on.

Hence anniversary $ell in L$ contributes at best $d(k-1)$ incidences with its low points, giving a absolute addition of at best $d(k-1)lvert Lrvert $ to the sum. If x is not a low point on $ell $ , afresh there are some $z_1 < dotsb < z_{k-1} < x$ in $ell $ , with anniversary one a adolescent of the abutting one. As L is a $K_{k,k}$ -free arrangement, amid the sets $ell in L$ there are at best $k-1$ absolute all these points. By the analogue of $P”$ , for anniversary $x in P”$ there are at best $m^{k-1}$ choices for such tuples $(z_1, dotsc , z_{k-1})$ . Appropriately x is adventure to at best $(k-1)m^{k-1}$ sets $ell in L$ for which it is not low, and the absolute cardinal of contributions of incidences in this case is at best $(k-1) m^{k-1} lvert P”rvert $ , so the affirmation follows.

Now we prove Acceptance 4.3. Let t be a accustomed cardinal to be called after and $m>0$ be addition constant to be called later. Ascertain the subsets

of P by defining $P_0 := P$ , and for anniversary $i=0,dots ,t-1$ , defining $P_{i 1}$ to be the set of credibility in $P_i$ that accept a $(k-1)$ -descendant with added than m accouchement in $(P_i, <)$ . By Antecedent 4.4, we have

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

Proof. By analogue of $P_t$ there is some $(k-1)$ -descendant $x’ in P_{t-1}$ of x which has at atomic $ m$ accouchement in $P_{t-1}$ . Let $S_{t-1} subseteq P_{t-1}$ denote the set of accouchement of $x’$ , so $lvert S_{t-1}rvert geq m$ . By about-face consecration for $i= t-1, t-2, dotsc , 0$ , we accept sets $S_{i} subseteq P_i$ of birth of x so that $lvert S_{i-1}rvert geq frac {lvert S_irvert m }{k d^k}$ . Afresh $lvert S_0rvert geq frac { m^t}{left (k d^kright )^{t-1}} $ , as wanted.

Let $S_i$ be given. By the analogue of $P_{i}$ and the assort principle, there is some $0 leq s leq k-1$ and $S^{prime }_i subseteq S_i$ such that $left lvert S^{prime }_iright rvert geq frac {lvert S_irvert }{k}$ and every $y in S^{prime }_i$ has a austere s-descendant $z_y in P_{i-1}$ with at atomic m accouchement in $P_{i-1}$ . Fix a aisle $I_y$ of breadth s abutting y to $z_y$ , and for $0 leq r leq s$ let $z^r_y$ denote the rth aspect on the aisle $I_y$ (so $z_y^0 = y $ , $z_y^s = z_y$ and $z_y^{r 1}$ is a adolescent of $z_y^r$ ). Ascertain $I^r := left { z^r_y : y in S^{prime }_iright }$ , so $I^0 = S^{prime }_i$ . Afresh $left lvert I^{r 1}right rvert geq frac {lvert I^rrvert }{d}$ (otherwise there is some aspect $z in I^{r 1}$ which has at atomic $d 1$ altered parents in $I^r$ , which would afresh anatomy an antichain of admeasurement $d 1$ , contradicting the bounded d-linearity of P). Hence

Now by antecedent every aspect in $I^s$ has at atomic m accouchement in $P_{i-1}$ ; denote the set of all the accouchement of the elements in $I^s$ by $S_{i-1} subseteq P_{i-1}$ . Then, afresh by d-linearity, $lvert S_{i-1}rvert geq frac {lvert I^srvert m}{d} geq frac {lvert S_irvert m }{k d^k}$ .

Thus if we accept $m, t$ such that

then we will get a contradiction, unless $P_t$ is empty. We conclude, for such m and t, that

If we booty $m := left ( frac {c log left (100 n_1right )}{log log left (100 n_1right )} right )^{frac {1}{k-1}}$ and t to be the accumulation allotment of $frac {c log left (100 n_1right )}{log log left (100 n_1right )}$ , and accept that c is abundantly ample about to k and d, afresh the affirmation follows.

Note that if two dyadic intervals intersect, afresh one charge be independent in the other.

Proof. Suppose that we accept some nested dyadic rectangles $D_1 supseteq D_2 supseteq dotsb supseteq D_k$ in R. As the accident blueprint is $K_{k,k}$ -free by hypothesis, $D_k$ may accommodate at best $(k-1)$ credibility from P. Removing all such rectangles repeatedly, we lose alone $(k-1) n_2$ incidences, and appropriately may accept that any nested adjustment in R is of breadth at best $k-1$ . In particular, any rectangle can be afresh at best $k-1$ times in R. Then, possibly accretion the cardinal of incidences by a assorted of $(k-1)$ , we may accept that there are no repetitions in R.

We now ascertain a affiliation $leq $ on R by declaring $I times J leq I’ times J’$ if $I subseteq I’$ and $J supseteq J’$ . This is a locally $(k-1)$ -linear fractional adjustment (by the antecedent paragraph: antisymmetry holds, as there are no repetitions in R, and application the actuality that all rectangles are dyadic, any antichain of admeasurement k central an breach would accord a nested adjustment of rectangles of breadth k).

For anniversary point p in P, let $ell _p$ be a subset of R consisting of all those rectangles in R that accommodate p; afresh $ell _p$ is a $(k-1)$ -linear set (again, any antichain gives a nested adjustment of rectangles of the aforementioned length). Finally, $p in R iff R in ell _p$ , appropriately the accumulating $left { ell _p : p in P right }$ is a $K_{k,k}$ -free adjustment and the affirmation now follows from Acceptance 4.3 with $d := k-1$ .

In this area we access a stronger apprenticed in Acceptance 2.17 (without the logarithmic factor) beneath a stronger acceptance that the accomplished semilinear affiliation X is $K_{k, dotsc ,k}$ -free (Corollary 5.12). And we appearance that if this stronger apprenticed does not authority for a accustomed semialgebraic relation, afresh the acreage operations can be recovered from this affiliation (see Aftereffect 5.14 for the absolute statement). These after-effects are deduced in Area 5.2 from a added accepted model-theoretic acceptance accepted in Area 5.1.

We anamnesis some accepted model-theoretic characters and definitions, and accredit to [Reference Marker15] for a accepted accession to archetypal approach and [Reference Berenstein and Vassiliev3] for added capacity on geometric structures.

Recall that $operatorname {acl}$ denotes the algebraic cease abettor – that is, if $mathcal {M} = (M, dotsc )$ is a first-order structure, $A subseteq M$ and a is a apprenticed tuple in M, afresh $a in operatorname {acl}(A)$ if it belongs to some apprenticed A-definable subset of $M^{lvert arvert }$ (this generalises the beeline amount in agent spaces and algebraic cease in fields). Throughout this section, we chase the accepted model-theoretic notation: depending on the context, autograph $BC$ denotes either the abutment of two subsets $B,C$ of M or the tuple acquired by concatenating the (possibly infinite) tuples $B,C$ of elements of M.

Definition 5.1. A complete first-order approach T in a accent $mathcal {L}$ is geometric if for any archetypal $mathcal {M} = (M, dotsc ) models T$ we accept the following:

1. The algebraic cease in $mathcal {M}$ satisfies the barter principle:

if $a,b$ are singletons in $mathcal {M}$ , $A subseteq M$ and $b in operatorname {acl}(A,a) setminus operatorname {acl}(A)$ , afresh $a in operatorname {acl} (A,b)$ .

2. T eliminates the $exists ^{infty }$ quantifier:

for every $mathcal {L}$ -formula $varphi (x,y)$ with x a distinct capricious and y a tuple of variables, there exists some $k in mathbb {N}$ such that for every $b in M^{lvert yrvert }$ , if $varphi (x,b)$ has added than k solutions in M, afresh it has always abounding solutions in M.

In models of a geometric theory, the algebraic cease abettor $operatorname {acl}$ gives acceleration to a matroid, and accustomed a a apprenticed tuple in M and $A subseteq M$ , $dim (a/A)$ is the basal cardinality of a subtuple $a’$ of a such that $operatorname {acl}(a cup A) = operatorname {acl}(a’ cup A)$ (in an algebraically bankrupt field, this is aloof the arete amount of a over the acreage generated by A). Finally, accustomed a apprenticed tuple a and sets $C,B subseteq M$ , we address to denote that $dim left (a/BC right ) = dim left (a / C right )$ .

The afterward acreage expresses that the matroid authentic by the algebraic cease is linear, in the faculty that the cease abettor behaves added like the amount in agent spaces, as against to algebraic cease in fields:

Recall that a linearly ordered anatomy $mathcal {M}=(M,<, dotsc )$ is o-minimal if every apprenticed subset of M is a apprenticed abutment of intervals (see, e.g., [Reference Van den Dries23]).

An o-minimal anatomy is beeline (i.e., any accustomed interpretable ancestors of even curves in T has ambit $leq 1$ ) if and alone if it is abominably locally modular.

In particular, every approach of an ordered agent amplitude over an ordered analysis arena is abominably locally modular (so Acceptance 5.6 applies to semilinear relations).

The afterward is a key model-theoretic lemma:

Proof. Accept the antecedent is untrue; afresh there abide some $(a_1, dotsc , a_r)$ in $mathcal {M}$ such that $(a_1,dotsc , a_r) in E$ , but $a_i notin operatorname {acl} left ( a_{neq i}, b right )$ for every $i in [r]$ , area $a_{neq i} := left {a_j : j in [r] setminus {i} right }$ .

By anemic bounded modularity, for anniversary $i in [r]$ there exists some baby set $C_i subseteq mathcal {M}$ such that

By addendum of , we may accept that for all $i in [r]$ . Appropriately by transitivity, , area $C := bigcup _{i in [r]} C_i$ .

Set $D := bigcap _{i in [r]} operatorname {acl} left ( a_{neq i}, b, Cright )$ .

Proof. Fix $i in [r]$ . As

and

, by agreement and transitivity we have

Note that $operatorname {acl}(a_i,C_i) subseteq operatorname {acl}left (a_{neq j},Cright )$ for every $i neq j in [r]$ , and appropriately $operatorname {acl}(a_i,C_i) cap operatorname {acl}(a_{neq i},b,C_i) subseteq D$ , and acutely $D subseteq operatorname {acl}(a_{neq i},b,C)$ . Appropriately , and in accurate .

By consecration we will accept sequences of tuples $bar {alpha }_i = left (a_i^{t}right )_{t in mathbb {N}}, i in [r]$ , in $mathcal {M}$ such that for every $i in [r]$ we have:

1. $a^t_i equiv _{D bar {alpha }_{<i} a_{>i}} a_i$ for all $t in mathbb {N}$ ;

2. $a^t_i neq a_i^{s}$ (as tuples) for all $s neq t in mathbb {N}$ ;

3. .

Fix $i in [r]$ and accept that we already chose some sequences $bar {a}_j$ for $1 leq j < i$ acceptable (1)–(3).

Claim C. We accept .

Proof. If $i=1$ , this affirmation becomes

, appropriately holds by Affirmation (A). So accept $i geq 2$ . We will appearance by consecration that for anniversary $l = 1, dotsc , i-1$ we have

For $l =1$ this is agnate to

, which holds by (3) for $i-1$ . So we accept this holds for $l < i-1$ – that is, we have

– and appearance it for $l 1$ . By acceptance and transitivity, we have

Also,

by (3) for $i-(l 1) < i$ . Afresh by transitivity again,

, which concludes the anterior step.

In particular, for $l = i-1$ we get – that is, . By transitivity and Affirmation (A), this implies , and we achieve by symmetry.

Using Affirmation (C) and addendum of , we can accept a adjustment $bar {alpha }_i = left (a^t_iright )_{t in mathbb {N}}$ so that $a^t_i equiv _{D bar {alpha }_{<i} a_{>i}} a_i$ and for every $t in mathbb {N}$ . By Affirmation (B) we accept $a_i notin operatorname {acl}(D)$ , appropriately $a_i^t notin operatorname {acl}(D)$ , appropriately $a^t_i notin operatorname {acl}left (bar {alpha }_{<i}, a_{>i}, a_i^{<t} right )$ , so in accurate all the tuples $left (a^t_iright )_{t in mathbb {N}}$ are pairwise-distinct and $bar {alpha }_i$ satisfies (1) and (2). By agreement and transitivity of , we get . This concludes the anterior footfall in the architecture of the sequences.

Finally, as (1) holds for all $i in [r]$ and b is independent in D, it follows that $left (a^{t_1}_{1}, dotsc , a^{t_r}_rright ) equiv _{b} (a_1, dotsc , a_r)$ , and appropriately $left (a^{t_1}_{1}, dotsc , a^{t_r}_rright ) in E$ for every $(t_1, dotsc , t_r) in mathbb {N}^r$ . By (1), anniversary of the sets $left {a^{t}_i : t in mathbb {N} right }, i in [r]$ , is absolute – contradicting the acceptance on E. This concludes the affidavit of the lemma.

Theorem 5.6. Accept that T is a geometric, abominably locally modular theory, and $mathcal {M} models T$ . Accept that $r in mathbb {N}_{geq 2}$ and $varphi (bar {x}_1, dotsc , bar {x}_r,bar {y})$ is an $mathcal {L}$ -formula after parameters, with $lvert bar {x}_irvert = d_i, lvert bar {y}rvert = e$ . Afresh there exists some $alpha = alpha (varphi ) in mathbb {R}_{>0}$ acceptable the following:

Given $b in M^{e}$ , accede the r-ary relation

Then for every $b in M^e$ , absolutely one of the afterward two cases charge occur:

1. $E_b$ is not $K_{k, dotsc , k}$ -free for any $k in mathbb {N}$ , or

2. for any apprenticed r-grid $B subseteq prod _{i in [r]} M^{d_i}$ , we have

Proof. Accept that $mathcal {N} = (N, dotsc )$ is an elementary addendum of $mathcal {M}$ and $b in M^{e}$ . Afresh for a anchored $k in mathbb {N}$ ,

is $K_{k, dotsc , k}$ -free if and alone if

is $K_{k, dotsc , k}$ -free, as this can be bidding by a first-order blueprint $psi (y)$ and $mathcal {M} models psi (b) iff mathcal {N} models psi (b)$ . Similarly, for a anchored $alpha in mathbb {R}$ , $lvert E_b cap Brvert leq alpha delta ^r_{r-1}(B)$ for every apprenticed r-grid $B subseteq prod _{i in [r]} M^{d_i}$ if and alone if $left lvert E^{prime }_b cap Bright rvert leq alpha delta ^r_{r-1}(B)$ for every apprenticed r-grid $B subseteq prod _{i in [r]} N^{d_i}$ (as for all anchored sizes of $B_1, dotsc , B_r$ , this action can be bidding by a first-order formula). Hence, casual to an elementary extension, we may accept that $mathcal {M}$ is $aleph _1$ -saturated.

As T eliminates $exists ^{infty }$ , there exists some $m = m(varphi ) in mathbb {N}$ such that for any $i in [r]$ , $b in M^e$ and tuple $bar {a} := left ( a_j in M^{d_j} : j in [r] setminus {i} right )$ , the fibre

is apprenticed if and alone if it has admeasurement $leq m$ .

Given an approximate $b in M^{e}$ such that $E_b$ is $K_{k, dotsc , k}$ -free, Antecedent 5.5 and bendability betoken that for every tuple $(a_1, dotsc , a_r) in E_b$ , there exists some $i in [r]$ such that the fibre $E^i_{bar {a}; b}$ is finite, appropriately $left lvert E^i_{bar {a}; b}right rvert leq m$ . This calmly implies that for any apprenticed r-grid $B subseteq prod _{i in [r]} M^{d_i}$ , we accept $lvert E_bcap Brvert leq m delta ^r_{r-1}(B)$ .

Restricting to distal structures, we can relax the acceptance ‘ $E_b$ is $K_{k, dotsc , k}$ -free for some k’ to ‘ $E_b$ does not accommodate a absolute artefact of absolute sets’ in Acceptance 5.6 (we accredit to, e.g., the accession in [Reference Chernikov and Starchenko6] or [Reference Chernikov, Galvin and Starchenko4] for a accepted altercation of model-theoretic distality and its access to combinatorics).

Now we appearance that in the o-minimal case, this aftereffect absolutely characterises anemic bounded modularity. By the trichotomy acceptance in o-minimal structures [Reference Peterzil and Starchenko18], we accept the afterward equivalence:

Proof. As every o-minimal anatomy is distal and every semilinear affiliation is apprenticed in an ordered agent amplitude over $mathbb {R}$ which is o-minimal and locally modular (Example 5.4), the aftereffect follows by Aftereffect 5.8.

We anamnesis the afterward appropriate case of the trichotomy acceptance in o-minimal structures belted to semialgebraic relations:

Using this fact, we accept the afterward added absolute alternative of Aftereffect 5.11 in the semialgebraic case:

Corollary 5.14. Let $X subseteq mathbb {R}^d$ be a semialgebraic set, and accede the first-order anatomy $mathcal {M} = (mathbb {R}, <, ,X)$ . Afresh the afterward are equivalent:

1. For any $r in mathbb {N}$ and any r-ary affiliation $Y subseteq prod _{i in [r]}mathbb {R}^{d_i}$ not absolute an r-grid $A = prod _{i in [r]}A_i$ with anniversary $A_i subseteq mathbb {R}^{d_i}$ infinite, there exists some $alpha in mathbb {R}$ such that $lvert Y cap Brvert leq alpha delta ^r_{r-1}(B)$ for every apprenticed r-grid B.

2. For every $d_1,d_2 in mathbb {N}$ and $Y subseteq mathbb {R}^{d_1} times mathbb {R}^{d_2}$ apprenticed (with parameters) in $mathcal {M}$ , if Y is $K_{k,k}$ -free for some $k in mathbb {N}$ , afresh there abide some $beta < frac {4}{3}$ and $alpha $ such that for any n and $B_i subseteq mathbb {R}^{d_i}$ with $lvert B_irvert = n$ , we have

3. $times restriction _{[0,1]^2}$ is not apprenticed in $mathcal {M}$ .

Proof. (1) $Rightarrow $ (2) is obvious.

For (2) $Rightarrow $ (3), application $times restriction _{[0,1]^2}$ the $K_{2,2}$ -free point-line accident affiliation in $mathbb {R}^2$ is apprenticed belted to $[0,1]^2$ , and the accepted configurations witnessing the lower apprenticed in Szemerédi–Trotter can be scaled bottomward to the assemblage box.

For (3) $Rightarrow $ (1), accept that (1) does not authority in $(mathbb {R},<, ,X)$ . Afresh necessarily some Y apprenticed in $(mathbb {R},<, ,X)$ is not semilinear, by Aftereffect 5.12. By Actuality 5.13, if Y is not semilinear, afresh $times restriction _{[0,1]^2}$ is apprenticed in the anatomy $(mathbb {R},<, ,Y)$ , appropriately in $(mathbb {R},<, ,X)$ .

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