Extremal graphs without exponentially-small bicliques

Boris Bukh

Abstract.

The Turán problem asks for the largest number of edges in an $n$ -vertex graph not containing a fixed forbidden subgraph $F$ . We construct a new family of graphs not containing $K_{s,t}$ , for $t=C^{s}$ , with $\Omega(n^{2-1/s})$ edges matching the upper bound of Kövári, Sós and Turán.

2020 Mathematics Subject Classification:

05D40, 14N07

Supported in part by U.S. taxpayers through NSF CAREER grant DMS-1555149.

1. Introduction

The Turán problem.

Let $F$ be a fixed graph. The Turán problem asks for the value of $\operatorname{ex}(n,F)$ , the largest number of edges in an $n$ -vertex graph not containing a copy of $F$ as a subgraph. The classic theorem of Erdős and Stone [12] gives an asymptotic for $\operatorname{ex}(n,F)$ when $F$ is not bipartite.

For bipartite $F$ , much less is known. Even the simplest case when $F$ is a complete bipartite graph $K_{s,t}$ is open. Specifically, Kövári, Sós and Turán [19] proved that

\operatorname{ex}(n,K_{s,t})=O_{s,t}(n^{2-1/s}).

Obviously, we may reverse the roles of $s$ and $t$ to obtain $\operatorname{ex}(n,K_{s,t})=O_{s,t}(n^{2-1/t})$ , which is superior if $t<s$ . So, from now on we discuss only the case $t\geq s$ . Though the implicit constant in the big-Oh notation has been improved by Füredi [15], the Kövári–Sós–Turán bound remains the only upper bound on $\operatorname{ex}(n,K_{s,t})$ . Many researchers conjecture that the Kövári–Sós–Turán bound is tight (e.g., [19, p. 52], [11, p. 6], [14, p. 257], [16, Conjecture 2.24]). However, apart from the numerous results for $s=2$ and $s=3$ (see [16, Section 3] for a survey), there are only two constructions attaining the Kövári–Sós–Turán bound for general $s>3$ . The first is due to Alon, Rónyai and Szabó [2] who, improving on the previous construction by Kollár, Rónyai and Szabó [18], showed that

(1)

\operatorname{ex}(n,K_{s,t})=\Omega_{s}(n^{2-1/s})\qquad\text{if }t>(s-1)!.

The construction is a clever use of norms over finite fields. The second, more recent class of constructions originating from [4] uses random varieties. It has hitherto provided inferior dependence of $t$ on $s$ . For example, [4] obtains (1) only for $t\geq s^{4s}$ . The advantage of these constructions is their flexibility, see [9, 6, 20, 17, 7, 24] for some of their variations and applications.

In this work, we use a novel version of the random algebraic method to construct graphs that match the Kövári–Sós–Turán bound for $t$ that is only exponential in $s$ .

Theorem 1.

Let $s\geq 2$ . Then

\operatorname{ex}(n,K_{s,t})=\Omega_{s}(n^{2-1/s})\qquad\text{if }t>9^{s}\cdot s^{4s^{2/3}}.

The Zarankiewicz problem.

Closely related to the Turán problem for $K_{s,t}$ -free graphs is the problem of Zarankiewicz [25]. It is the asymmetric version of the Turán problem. It is well-known that in the study of the growth rate of $\operatorname{ex}(n,F)$ we may assume that the $n$ -vertex graph is bipartite. To distinguish the two parts of a bipartite graph, we shall call them the left and right. A copy of $K_{s,t}$ in a bipartite graph $G$ can be situated in two ways: either the $s$ vertices are in the left part, or the $s$ vertices are in the right part. In the Zarankiewicz problem, we forbid only the former case. So, we say that $G$ is a sided graph if it is bipartite with distinguished left and right parts. The Zarankiewicz problem then asks for the estimate on the number of edges in a sided graph not containing $K_{s,t}$ , which is regarded as a sided graph with $s$ vertices on the left and $t$ vertices on the right.

An important consequence of making the graph bipartite with distinguished parts is that the two parts can (possibly) be of very unequal size. This often occurs in applications (see e.g. [1, 23]). With this in mind, define $z(m,n;s,t)$ as the largest number of edges in a sided graph with $m$ vertices on the left and $n$ vertices on the right that contains no sided $K_{s,t}$ . The bound of Kövári, Sós and Turán for the Zarankiewicz problem takes the form

z(m,n;s,t)=O_{s,t}(mn^{1-1/s}).

In the symmetric case when $m=n$ , the best known constructions for Zarankiewicz problem are the same as the best bipartite constructions for the Turán problem. This is not so for our approach: we are able to take the advantage of the fact that only one orientation of $K_{s,t}$ is forbidden to obtain a lower bound on $z(n,n;s,t)$ that is superior to the corresponding bound for $\operatorname{ex}(n,K_{s,t})$ in Theorem 1.

Theorem 2.

a)

Suppose $s,t,m,n,k\geq 3$ are integers that satisfy the inequalities $\log_{n}m\leq\nobreak\frac{s^{k-2}}{k!\log^{2k}s}$ and $k\leq\frac{s}{2\log^{3}s}$ . Then

$z(m,n;s,t)=\Omega_{s}(mn^{1-1/s})\qquad\text{if }t>k^{s}\cdot e^{2s/\log s}.$

In particular, $z(n,n;s,t)=\Omega_{s}(n^{2-1/s})$ for $t>3^{s+o(s)}$ .
b)

For each $s\geq 3$ there is a constant $c_{s}>0$ such that if the inequality $\log_{n}m\leq c_{s}t^{\frac{1}{1+2\log s}}$ holds, then

$z(m,n;s,t)=\Omega_{s}(mn^{1-1/s}).$

Part (b) is an improvement on the result of Conlon [8], who proved the $\Omega_{s}(mn^{1-1/s})$ bound under the condition $\log_{n}m\leq c_{s}^{\prime}t^{1/(s-1)}$ . In the same article, Conlon asked if the bound holds for $\log_{n}m\leq t/s$ , which would be tight if true.

Main proof idea, in a nutshell.

The key novelty in our construction is that it is ‘bumpy’. All the previous algebraic constructions, whether random or not, were flat: the vertex of the graph was always an affine or a projective space over $\mathbb{F}_{q}$ , which was occasionally mildly mutilated by having a lower-dimensional subset removed, or stitched to itself via a quotient operation. These are finite field analogues of $\mathbb{R}^{n}$ with a flat metric. In contrast, the vertex set in our construction is a solution set to a family of random polynomial equations, and cannot be flattened with any change of coordinates.

Proof ideas, in more detail.

To explain our construction, we recall the important ingredients in the previous random algebraic constructions of $K_{s,t}$ -free graphs. The key is an estimate on the probability that the variety $\mathbf{V}(f_{1},\dotsc,f_{s})$ cut out by $s$ random polynomials contains many points. This relies on two inputs. The first is Bezout’s theorem, which is used to conclude that if $\mathbf{V}(f_{1},\dotsc,f_{s})$ is zero-dimensional, then it contains at most $\prod\deg f_{i}$ many points. The second input is a bound on the probability that $\mathbf{V}(f_{1},\dotsc,f_{s})$ is of codimension less than $s$ .

To prove that bound, the original paper [4] uses Hilbert functions (though it did not use probabilistic language). However, almost all the subsequent works use the approach from [5] relying on the Lang–Weil bounds, the only exception being an elegant argument in [8] which can be described as an implicit use of Hilbert functions.

In this paper, we go back to the explicit use of Hilbert functions. We show that the codimension of $\mathbf{V}(f_{1},\dotsc,f_{s})$ is extremely likely equal to $s$ , unless $s$ is close to the dimension of the ambient space. That is precisely the situation in the previous works, where the graph’s vertex set was $s$ -dimensional set $\mathbb{F}_{q}^{s}$ . To bypass this obstacle we construct the vertex set in two steps: At the start we use affine space $\mathbb{F}_{q}^{s+r}$ of slightly larger dimension. This way the varieties of the form $\mathbf{V}(f_{1},\dotsc,f_{s})$ that we obtain have dimension $r$ with very high probability. We then shrink the vertex set to a random subvariety of codimension $r$ , thereby cutting all the varieties of the form $\mathbf{V}(f_{1},\dotsc,f_{s})$ at once.

Our second innovation concerns the need to control $\prod\deg f_{i}$ in the Bezout’s bound. In the construction of Turán graphs, the random polynomials $f_{1}(y),\dotsc,f_{s}(y)$ arise as specializations of a single polynomial $g(x,y)$ in two sets of variables. A simple way to ensure that the random polynomials $f_{1}(y),\dotsc,f_{s}(y)$ are mutually independent is to make $g$ a random polynomial of degree at least $s$ . That makes $\prod\deg f_{i}$ grow like $s^{s}$ . To circumvent this, we further replace the $(s+r)$ -dimensional affine space by a variety $U$ that has the property that every specialization of a random polynomial $g$ of bounded degree to any $s$ points of $U$ yields $s$ mutually independent polynomials. We call these $m$ -independent varieties. The construction of $m$ -independent varieties occupies the bulk of the paper (Sections 4 and 5).

Finally, we want to highlight an auxiliary contribution of this work that is of independent interest. The construction of $m$ -independent varieties depends on a bound on the number of minimal linear dependencies between $m$ ’th powers of linear forms, i.e., $0=\sum\alpha_{i}\ell_{i}(x)^{m}$ where $\ell_{i}(x)=c_{i,0}x_{0}+\dotsb+c_{i,b}x_{b}$ . Here, ‘minimal’ means that no proper subset of the $m$ ’th powers of these linear forms is linearly dependent. Representation of polynomials by sums of $m$ ’th powers of linear forms has been much studied, motivated primarily by the Waring problem. In particular, we adapt the argument which is implicit in the work of Białynicki-Birula and Schinzel [3] to our purposes. Though the argument in [3] suffices to obtain an exponential bound in Theorem 1, we go beyond and obtain a stronger bound that yields the smaller exponent base of $9$ in Theorem 1.

Paper organization.

We begin by collecting the algebraic tools we require in Section 2. The concept of an $m$ -independent set, which is central to the proof of Theorem 1, is introduced in Section 3. The $m$ -independent varieties used in the proof of Theorem 1 are constructed in Sections 4 and 5. Finally, in Sections 6 and 7 we prove Theorems 1 and 2 respectively.

Acknowledgments.

I am thankful to Chris Cox for extensive feedback on an earlier version of this paper. I am grateful to Jacob Tsimerman for discussions on numerous topics related to this paper, and especially for motivating me to prove Lemma 22 in its current form, and to Anamay Tengse, Mrinal Kumar, Ramprasad Saptharishi for spotting a serious error in the proof of Lemma 22 in the previous version of this paper. Finally, I am grateful to the anonymous referees for a number of constructive comments.

2. Algebraic tools

To make this paper maximally accessible, we tried to keep the use of algebra to the minimum. In particular, we use counting arguments even when similar algebraic arguments could have provided slightly superior numeric constants. Despite this, we require basic familiarity with algebraic geometry on the level of the first chapter of Shafarevich’s book [21]. We collect the other algebraic tools in this section.

Varieties and their $\mathbb{F}_{q}$ -points.

The integer $q$ will denote a prime power. We shall work exclusively with fields $\mathbb{F}_{q}$ and $\overline{\mathbb{F}_{q}}$ . All varieties in this paper are quasi-projective over the field $\overline{\mathbb{F}_{q}}$ . We write $\mathbf{V}(f_{1},\dotsc,f_{t})$ for the projective variety cut out by homogeneous polynomials $f_{1},\dotsc,f_{t}$ . We denote the vector space of homogeneous polynomials of degree $m$ in $b+1$ variables with coefficients in $\mathbb{F}_{q}$ by $\mathbb{F}_{q}[x_{0},\dotsc,x_{b}]_{m}$ . We also work with products of projective spaces $\mathbb{P}^{a}\times\mathbb{P}^{b}$ . The set of bihomogeneous polynomials of bidegrees $(m,m^{\prime})$ on $\mathbb{P}^{a}\times\mathbb{P}^{b}$ is denoted by $\mathbb{F}_{q}[x_{0},\dotsc,x_{a}]_{m}\otimes\nobreak\mathbb{F}_{q}[y_{0},\dotsc,y_{b}]_{m^{\prime}}$ .

We write monomials using the multiindex notation: for a multiindex $\beta=(\beta_{0},\beta_{1},\dotsc,\beta_{b})\in\mathbb{Z}_{\geq 0}^{b+1}$ , the notation $x^{\beta}$ stands for the monomial $x_{0}^{\beta_{0}}x_{1}^{\beta_{1}}\dotsb x_{b}^{\beta_{b}}$ . A general homogeneous polynomial of degree $m$ is thus written as $\sum_{\lvert\beta\rvert=m}c_{\beta}x^{\beta}$ .

The graphs we shall construct in the proofs of Theorems 1 and 2 will consist of the $\mathbb{F}_{q}$ -points of certain varieties. We denote the $\mathbb{F}_{q}$ -points of a variety $V\subseteq\mathbb{P}^{b}$ by $V(\mathbb{F}_{q})$ or (if the variety $V$ is a complicated expression) by $V\cap\mathbb{P}^{b}(\mathbb{F}_{q})$ . We shall use the following bounds on the number of $\mathbb{F}_{q}$ -points.

Lemma 3 (Weakening of [10, Corollary 3.3]).

Suppose $V\subseteq\mathbb{P}^{b}$ is any $k$ -dimensional variety of degree $d$ . Then $\lvert V(\mathbb{F}_{q})\rvert\leq d\lvert\mathbb{P}^{k}(\mathbb{F}_{q})\rvert$ .

Lemma 4.

a)

Let $m_{1},\dotsc,m_{r}$ be positive integers, and let $Y\subseteq\mathbb{P}^{b}(\mathbb{F}_{q})$ be a non-empty set. Suppose that $g_{1},\dotsc,g_{r}\in\mathbb{F}_{q}[x_{0},\dotsc,x_{b}]$ are random homogeneous polynomials of degrees $\deg g_{i}=m_{i}$ . Then

(2) $\Pr\left[\lvert Y\cap\mathbf{V}(g_{1},\dotsc,g_{r})\rvert\leq\frac{\lvert Y\rvert}{2q^{r}}\right]\leq\frac{4q^{r}}{\lvert Y\rvert}.$
b)

The same holds for bihomogeneous polynomials, i.e., if $Y\subseteq\mathbb{P}^{a}(\mathbb{F}_{q})\times\mathbb{P}^{b}(\mathbb{F}_{q})$ is a non-empty set and $g_{1},\dotsc,g_{r}$ are random bihomogeneous polynomials of bidegrees $\deg g_{i}=(m_{i},m_{i}^{\prime})$ with $m_{i},m_{i}^{\prime}\geq\nobreak 1$ , then (2) holds.

Proof.

For a point $y\in Y$ , let $R_{y}$ be the indicator random variable of the event $y\in\mathbf{V}(g_{1},\dotsc,g_{r})$ . Let $y,y^{\prime}\in Y$ be two distinct points. We claim that $\mathbb{E}[R_{y}]=\mathbb{E}[R_{y^{\prime}}]=1/q^{r}$ and that the random variables $R_{y}$ and $R_{y^{\prime}}$ are independent. To see this in the case (a), apply a change of coordinates so that $y=[1:0:0:\dotsc:0]$ and $y^{\prime}=[0:1:0:\dotsc:0]$ . The polynomial $g_{i}$ vanishes at $y$ if and only if the coefficient of $x_{0}^{m}$ vanishes. Similarly, $g_{i}(y^{\prime})=0$ if and only if the coefficient of $x_{1}^{m}$ vanishes. In the case (b), write $y=(y_{a},y_{b})$ and $y^{\prime}=(y_{a}^{\prime},y_{b}^{\prime})$ with $y_{a},y_{a}^{\prime}\in\mathbb{P}^{a}(\mathbb{F}_{q})$ and $y_{b},y_{b}^{\prime}\in\mathbb{P}^{b}(\mathbb{F}_{q})$ . Since $y\neq y^{\prime}$ we may assume that $y_{b}\neq y_{b}^{\prime}$ (by swapping the roles of $\mathbb{P}^{a}$ and $\mathbb{P}^{b}$ if necessary). We can then change the coordinates on $\mathbb{P}^{b}$ in the same way as in the case (a), and observe that the vanishing of $g_{i}$ at $y$ and $y^{\prime}$ depends on disjoint sets of coefficients. This proves the claim.

Let $R=\sum_{y\in Y}R_{y}$ . From the pairwise independence of the $R_{y}$ ’s, it follows that

\operatorname{Var}[R]=\sum_{y\in Y}\operatorname{Var}[R_{y}]=\tfrac{1}{q^{r}}(1-\tfrac{1}{q^{r}})\lvert Y\rvert\leq\lvert Y\rvert/q^{r}.

Since $\mathbb{E}[R]=\sum_{y}\mathbb{E}[R_{y}]=\lvert Y\rvert/q^{r}$ , Chebyshev’s inequality then implies that

\Pr\bigl{[}\lvert R-\lvert Y\rvert/q^{r}\rvert\geq\lambda\sqrt{\lvert Y\rvert/q^{r}}\bigr{]}\leq\lambda^{-2},

and the lemma follows upon taking $\lambda=\sqrt{\lvert Y\rvert/4q^{r}}$ . ∎

Bézout’s inequality.

For a reducible variety $V\subseteq\mathbb{P}^{b}$ , define the total degree $\deg(V)$ to be the sum of the degrees of the irreducible components of $V$ .

Lemma 5 (Bézout’s inequality, [13, p. 228, Example 12.3.1]).

Let $V$ and $W$ be two varieties in $\mathbb{P}^{b}$ . Then

\deg(V\cap W)\leq\deg(V)\deg(W).

Hilbert functions.

For a homogeneous ideal $I$ , the Hilbert function $H_{I}(m)$ is defined as the codimension of $I_{m}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}I\cap\overline{\mathbb{F}_{q}}[x_{0},\dotsc,x_{b}]_{m}$ in $\overline{\mathbb{F}_{q}}[x_{0},\dotsc,x_{b}]_{m}$ . Equivalently, if $I=I(V)$ is the homogeneous ideal of polynomials vanishing on a variety $V$ , then $H_{I}(m)$ is the dimension of the subspace of functions on $V$ induced by all homogeneous polynomials of degree $m$ .

We shall use the following bound on the Hilbert function.

Lemma 6.

Let $V$ be a variety of dimension $k$ in $\mathbb{P}^{b}$ . Then its Hilbert function satisfies

H_{I(V)}(m)\geq\binom{m+k}{m}.

Though more general bounds are known (e.g. [22, Theorem 2.4]), we give a proof for completeness.

Proof.

By [21, Theorem 1.15 and Corollary 1.6] there is a linear projection $\pi\colon V\to\mathbb{P}^{k}$ , which is finite and surjective. Then the pullback of a homogeneous polynomial of degree $m$ on $\mathbb{P}^{k}$ is a homogeneous polynomial of degree $m$ on $V$ . The result follows since the space of homogeneous degree- $m$ polynomials on $\mathbb{P}^{k}$ is of dimension $\binom{m+k}{m}$ , and the pullback map has trivial kernel. ∎

We use Hilbert functions to bound the probability that a random polynomial vanishes on a given variety.

Lemma 7.

Let $V$ be any variety in $\mathbb{P}^{b}$ . Let $g\in\mathbb{F}_{q}[x_{0},\dotsc,x_{b}]_{m}$ be a random homogeneous degree- $m$ polynomial. Then the probability that $g$ vanishes on $V$ is

\Pr[g_{|V}=0]\leq q^{-H_{I(V)}(m)}.

Proof.

By the definition of the Hilbert function, $H_{I(V)}(m)$ is the codimension (over $\overline{\mathbb{F}_{q}}$ ) of $I(V)_{m}$ in $\overline{\mathbb{F}_{q}}[x_{0},\dotsc,x_{b}]_{m}$ . This implies that the codimension of $I(V)_{m}\cap\mathbb{F}_{q}[x_{0},\dotsc,x_{b}]$ in $\mathbb{F}_{q}[x_{0},\dotsc,x_{b}]_{m}$ is at least $H_{I(V)}(m)$ . Since $\Pr[g_{|V}=0]=\Pr[g\in I(V)_{m}\cap\mathbb{F}_{q}[x_{0},\dotsc,x_{b}]]$ , the lemma follows. ∎

3. Definition and uses of $m$ -independence

If $t$ is small compared to $b$ , then the Hilbert function of $t$ generic points in $\mathbb{P}^{b}$ is $H_{I(\{p_{1},\dotsc,p_{t}\})}(m)=t$ for $m\geq 1$ . As we shall see shortly, the point sets satisfying $H_{I(\{p_{1},\dotsc,p_{t}\})}(m)=t$ behave random-like with respect to the degree- $m$ polynomials. Since this property will be key to our construction of Turán graphs, we focus on the sets lacking this pleasant property.

Definition 8.

We say that points $p_{1},\dotsc,p_{t}\in\mathbb{P}^{b}$ are $m$ -dependent if $H_{I(\{p_{1},\dotsc,p_{t}\})}(m)<t$ . We furthermore say that they are minimally $m$ -dependent if no proper subset of these points is $m$ -dependent.

Definition 9.

A set $X\subset\mathbb{P}^{b}$ is called $s$ -wise $m$ -independent if no $s$ distinct points of $V$ are $m$ -dependent.

Though we define $m$ -dependence via Hilbert functions, there are two other ways to think about the concept that will be useful. The first way is to think of a homogeneous degree- $m$ polynomial $f$ on $\mathbb{P}^{b}$ as a linear form in its coefficients. Specialization of $f$ to $f(p)$ gives distinct linear forms for distinct $p\in\mathbb{P}^{b}$ . It is easy to see from the definition of $m$ -dependence that the points $p_{1},\dotsc,p_{t}$ are $m$ -dependent if and only if $f(p_{1}),\dotsc,f(p_{t})$ are linearly dependent as linear forms in $\binom{b+m}{m}$ variables.

The second way to understand $m$ -dependence is via projective space duality, which we think of as an identification between points of $\mathbb{P}^{b}$ and linear forms. Namely, to each point $p\in\mathbb{P}^{b}$ we associate the linear form $\ell$ in $b+1$ variables defined by $\ell(x)\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\langle x,p\rangle$ . Because points in the projective space are defined only up to a multiplication by a non-zero scalar, this linear form too is defined up to a multiplication by a non-zero scalar.

Lemma 10.

Let $p_{1},\dotsc,p_{t}\in\mathbb{P}^{b}$ be any $t$ points, and let $\ell_{1}(x),\dotsc,\ell_{t}(x)$ be the linear forms associated to these points. The following are equivalent:

a)

points $p_{1},\dotsc,p_{t}$ are $m$ -dependent,
b)

there is a linear relation of the form

(3) $c_{1}\prod_{j=1}^{m}\ell_{1}(x_{j})+\dotsb+c_{t}\prod_{j=1}^{m}\ell_{t}(x_{j})=0.$

Furthermore, if the characteristic of $\mathbb{F}_{q}$ is at least $m$ , then the following condition is also equivalent to the two above:

c)

there is a linear relation of the form

(4) $c_{1}\ell_{1}(x)^{m}+\dotsb+c_{t}\ell_{t}(x)^{m}=0.$

Proof of (a) $\iff$ (b)..

Let $k_{1},\dotsc,k_{m}\in\{0,1,\dotsc,b\}$ be arbitrary, and let $\beta=(\beta_{0},\beta_{1},\dotsc,\beta_{b})$ be the tally vector, i.e., $\beta_{i}$ is the number of elements among $k_{1},\dotsc,k_{m}$ that are equal to $i$ . Then the coefficient of $x_{1,k_{1}}\dotsb x_{m,k_{m}}$ in $\prod_{j}\ell_{i}(x_{j})$ is equal to $p_{i}^{\beta}$ . Therefore, the linear relation in (3) holds if and only if $c_{1}p_{1}^{\beta}+\dotsb+c_{t}p_{t}^{\beta}=0$ for every $\beta\in\mathbb{Z}_{\geq 0}$ satisfying $\lvert\beta\rvert=m$ . As discussed in the paragraph following Definitions 8 and 9, the latter condition is equivalent to points $p_{1},\dotsc,p_{t}$ being $m$ -dependent.

Proof of (b) $\implies$ (c)..

Trivial, by setting $x_{1}=\dotsb=x_{m}$ .

Proof of (c) $\implies$ (b)..

Set $x=x_{1}+\dotsb+x_{m}$ in (4), and expand. Let $k_{1},\dotsc,k_{m}$ be arbitrary, and define $\beta$ as in the proof of (a) $\iff$ (b). The coefficient of $x_{1,k_{1}}\dotsb x_{m,k_{m}}$ in the expansion is $\binom{m}{\beta_{0},\beta_{1},\dotsc,\beta_{b}}$ times larger than the coefficient of the same monomial in (3). Since $\binom{m}{\beta_{0},\beta_{1},\dotsc,\beta_{b}}\neq 0$ follows from the assumption on the characteristic, the desired implication follows. ∎

If points $p_{1},\dotsc,p_{t}$ are not $m$ -dependent, we say that they are $m$ -independent.

The usefulness of these definitions comes from the combination of two simple observations:

Proposition 11.

Suppose that $v_{1},\dotsc,v_{s}\in\mathbb{P}^{a}(\mathbb{F}_{q})$ are $m$ -independent points. Pick a random polynomial $g\in\nobreak\mathbb{F}_{q}[x_{0},\dotsc,x_{a}]_{m}\otimes\mathbb{F}_{q}[y_{0},\dotsc,y_{b}]_{m^{\prime}}$ uniformly among all bihomogeneous polynomials of bidegree $(m,m^{\prime})$ on $\mathbb{P}^{a}\times\mathbb{P}^{b}$ . Then the $s$ random polynomials $g(v_{1},y),\dotsc,g(v_{s},y)\in\mathbb{F}_{q}[y_{0},\dotsc,y_{b}]_{m}$ are mutually independent.

Proof.

Write $g$ as $g(x,y)=\sum_{\lvert\beta\rvert=m^{\prime}}y^{\beta}g_{\beta}(x)$ . Since the coefficient of $y^{\beta}$ in $g(v_{i},y)$ is $g_{\beta}(v_{i})$ , it suffices to show that $g_{\beta}(v_{1}),\dotsc,g_{\beta}(v_{s})$ are independent for every $\beta$ .

Think of $g_{\beta}(v_{i})$ as a linear function of the coefficients of $g_{\beta}$ . By the alternative definition of $m$ -independence discussed above, the linear functions $g_{\beta}(v_{1}),\dotsc,g_{\beta}(v_{s})$ are linearly independent. Since the coefficients of $g_{\beta}$ are chosen independently and uniformly from $\mathbb{F}_{q}$ , this implies that the random variables $g_{\beta}(v_{1}),\dotsc,g_{\beta}(v_{s})$ are independent. ∎

Define the function

M_{k}(t)\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\min\Bigl{\{}m:\binom{m+k}{k}\geq t\Bigr{\}}.

Recall that a variety $W$ is said to be of pure dimension $k$ if all of its irreducible components are of dimension $k$ .

Proposition 12.

Let $0\leq s\leq k\leq b$ be integers, and let $W\subseteq\mathbb{P}^{b}$ be a variety of pure dimension $k$ and degree at most $D$ .

a)

If $h_{1},\dotsc,h_{s}$ are independent random homogeneous polynomials of degree $m$ in $b+1$ variables with $\mathbb{F}_{q}$ -coefficients, then

$\Pr\bigl{[}\dim(W\cap\mathbf{V}(h_{1},\dotsc,h_{s}))>k-s\bigr{]}\leq Cq^{-\binom{k-s+1+m}{m}},$

where the constant $C=C(k,m,D)>0$ depends only on $k,m$ and $D$ .
b)

Let $T$ and $\delta_{1},\dotsc,\delta_{s}$ be positive integers satisfying $\delta_{i}\geq M_{k-i+1}(T)$ . If $h_{1},\dotsc,h_{s}$ are independent random homogeneous polynomials of degrees $\deg h_{i}=\delta_{i}$ in $b+1$ variables with $\mathbb{F}_{q}$ -coefficients, then

$\Pr\bigl{[}\dim(W\cap\mathbf{V}(h_{1},\dotsc,h_{s}))>k-s\bigr{]}\leq Cq^{-T},$

where the constant $C=C(T,s,D,\delta_{1},\dotsc,\delta_{s})>0$ depends only on $T,s$ , $D$ and on the polynomial degrees $\delta_{1},\dotsc,\delta_{s}$ .

Proof.

Part (a) is the special case of part (b) with $T=\binom{k-s+1+m}{m}$ and $\delta_{1}=\dotsb=\delta_{s}=m$ . So, it suffices to prove part (b).

The proof is by induction on $s$ . Let $\mathcal{U}$ be the set of all irreducible components of $W\cap\mathbf{V}(h_{1},\dotsc,h_{s-1})$ . Since $\mathbf{V}(h_{1},\dotsc,h_{s-1})$ is cut out by $s-1$ polynomials and $W$ is of pure dimension $k$ , it follows that $\dim(U)\geq k-s+1$ for each $U\in\mathcal{U}$ [21, Corollary 1.14]. So, from Lemmas 6 and 7 we see that, for any fixed $U\in\mathcal{U}$ ,

\Pr[h_{s}\text{ vanishes on }U]\leq q^{-\binom{k-s+1+\delta_{s}}{k-s+1}}\leq q^{-\binom{k-s+1+M_{k-s+1}(T)}{k-s+1}}\leq q^{-T}.

Bézout’s inequality tells us that $\lvert\mathcal{U}\rvert\leq D\prod_{i=1}^{s-1}\delta_{i}$ , and hence the probability that $h_{s}$ vanishes on some component of $W\cap V(h_{1},\dotsc,h_{s-1})$ is at most $D\prod_{i=1}^{s-1}\delta_{i}\cdot q^{-T}$ . By the induction hypothesis, the variety $W\cap\mathbf{V}(h_{1},\dotsc,h_{s-1})$ is of dimension exceeding $k-s+1$ with probability at most $C(T,s-\nobreak 1,D,\delta_{1},\dotsc,\delta_{s-1})q^{-T}$ , and so the probability that the variety $W\cap\mathbf{V}(h_{1},\dotsc,h_{s-1},h_{s})$ is of dimension exceeding $k-s$ is at most

D\prod_{i=1}^{s-1}\delta_{i}\cdot q^{-T}+C(t,s-1,D,\delta_{1},\dotsc,\delta_{s-1})q^{-T},

which is at most $C(T,s,D,\delta_{1},\dotsc,\delta_{s})q^{-T}$ for a suitable $C(\dotsb)$ . ∎

Combining these two observations we obtain the following handy result.

Lemma 13.

Let $0\leq s\leq k\leq b$ be integers. Suppose that $W\subseteq\mathbb{P}^{b}$ is a variety of pure dimension $k$ , and $X\subset\mathbb{P}^{a}(\mathbb{F}_{q})$ is an $s$ -wise $m$ -independent set of size $\lvert X\rvert\leq c^{\prime}q^{\frac{1}{s}\binom{k-s+1+m}{m}}$ where $c^{\prime}=c^{\prime}(m,s,\deg W)>0$ is sufficiently small. Let $g\in\mathbb{F}_{q}[x_{0},\dotsc,x_{a}]_{m}\otimes\mathbb{F}_{q}[y_{0},\dotsc,y_{b}]_{m}$ be a random bihomogeneous polynomial of bidegree $(m,m)$ on $\mathbb{P}^{a}\times\mathbb{P}^{b}$ . Then the following holds with probability at least $\tfrac{4}{5}$ : For every $s$ distinct points $v_{1},\dotsc,v_{s}\in X$ the variety

\{w\in W:g(v_{1},w)=\dotsb=g(v_{s},w)=0\}

is of dimension $k-s$ .

Proof.

By Proposition 11, polynomials $g(v_{1},w),\dotsc,g(v_{s},w)$ are mutually independent, for every $s$ distinct points $v_{1},\dotsc,v_{s}\in X$ . By Item 12(a) and the union bound over all $(v_{1},\dotsc,v_{s})\in X^{s}$ , the probability that $g$ does not satisfy the lemma is at most $\lvert X\rvert^{s}\cdot C(m,s,\deg W)q^{-\binom{k-s+1+m}{m}}\leq(c^{\prime})^{s}C(m,s,\deg W)$ . So, choosing $c^{\prime}$ small enough works. ∎

4. Construction of $m$ -independent varieties

For positive integers $b,m,t$ , let

	$\displaystyle\Phi_{t}(b,m)$	$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{(p_{1},\dotsc,p_{t})\in(\mathbb{P}^{b})^{t}:p_{1},\dotsc,p_{t}\text{ are minimally $m$-dependent}\},$
	$\displaystyle\phi_{t}(b,m)$	$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\dim\Phi_{t}(b,m).$

Observe that $\Phi_{t}(b,m)$ is a variety. Indeed, for a subset $I\subseteq[t]\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{1,2,\dotsc,t\}$ , define

\widetilde{\Phi}_{I}(b,m)\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{(p_{1},\dotsc,p_{t})\in(\mathbb{P}^{b})^{t}:(p_{i})_{i\in I}\text{ are $m$-dependent}\}.

The set $\widetilde{\Phi}_{I}(b,m)$ is a projective variety because of the equivalence of $m$ -dependence of points and linear dependence of the linear $m$ -th powers of respective linear forms, which we discussed above. From this, it follows that $\Phi_{t}(b,m)=\widetilde{\Phi}_{[t]}(b,m)\setminus\bigcup_{I\subsetneq[t]}\widetilde{\Phi}_{I}(b,m)$ is a difference between two projective varieties, and so itself is a (quasiprojective) variety.

We shall upper bound the functions $\phi_{t}$ in the next section. But first we show how to use these bounds to construct $m$ -independent varieties.

Lemma 14.

Suppose that $b,m,Z,s\geq 1$ are integers satisfying

(5)

Z>\frac{1}{t-1}\phi_{t}(b,m)\qquad\text{ for all }t=2,3,\dotsc,s.

Let $f_{1},\dotsc,f_{Z}$ be generic degree- $m$ homogeneous polynomials in $b+1$ variables. Then the variety $\mathbf{V}(f_{1},\dotsc,f_{Z})$ is $s$ -wise $m$ -independent.

As we shall show in Item 22(a), $\Phi_{t}(b,m)=\emptyset$ for $t\leq m+1$ . Hence, the condition (5) is non-vacuous only for $t=m+2,m+3,\dotsc,s$ .

Proof of Lemma 14.

It suffices to show that $\mathbf{V}(f_{1},\dotsc,f_{Z})$ contains no set of $t$ minimally $m$ -dependent points for every $t=2,3,\dotsc,s$ . Fix $t$ in this range.

Note that whenever points $p_{1},\dotsc,p_{t}\in\mathbb{P}^{b}$ are minimally $m$ -dependent, their Hilbert function satisfies $H_{I(\{p_{1},\dotsc,p_{t}\})}(m)=t-1$ . Indeed, the inequality $H_{I(\{p_{1},\dotsc,p_{t}\})}(m)\leq t-1$ follows from the definition of $m$ -dependence, and $H_{I(\{p_{1},\dotsc,p_{t}\})}(m)\geq t-1$ follows from minimality. This means that the vector space $I(\{p_{1},\dotsc,p_{t}\})_{m}$ is of codimension $t-1$ in $\overline{\mathbb{F}_{q}}[x_{0},\dotsc,x_{b}]_{m}$ , and hence $I(\{p_{1},\dotsc,p_{t}\})_{m}^{Z}$ is of codimension $Z(t-1)$ in $(\overline{\mathbb{F}_{q}}[x_{0},\dotsc,x_{b}]_{m})^{Z}$ .

Since $Z(t-1)>\phi_{t}(b,m)$ , we can use the algebraic version of the union bound to deduce that for generic degree- $m$ homogeneous polynomials $f_{1},\dotsc,f_{Z}$ , the variety $\mathbf{V}(f_{1},\dotsc,f_{Z})$ does not contain any minimally $m$ -dependent set $\{p_{1},\dotsc,p_{t}\}$ . Indeed, write $F\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\overline{\mathbb{F}_{q}}[x_{0},\dotsc,x_{b}]_{m}^{Z}$ and regard it as a variety with polynomials’ coefficients as indeterminants. Define the variety

V\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{(f_{1},\dotsc,f_{Z},p_{1},\dotsc,p_{t})\in F\times\Phi_{t}(b,m):f_{i}(p_{j})=0\text{ for all }i,j\}.

Consider the projection of $V$ onto $F$ . Our claim is that the fiber of a generic $\vec{f}\in F$ is empty. If this is not so, then $\dim V\geq\dim F$ . However, for the projection of $V$ onto the $\Phi_{t}(b,m)$ factor, every fiber is of codimension $Z(t-1)$ , and so $\dim V\leq\dim\Phi_{t}(b,m)+(\dim F-Z(t-1))<\dim F$ , which is a contradiction, proving our claim. ∎

Lemma 15.

Suppose that $b,m,Z,s$ are integers satisfying the condition (5) and $b-Z\geq 1$ , and assume that $q\geq q_{0}(b,m,Z)$ is sufficiently large in terms of $b,m,Z$ . Then there exist degree- $m$ polynomials $f_{1},\dotsc,f_{Z}\in\mathbb{F}_{q}[x_{0},\dotsc,x_{b}]_{m}$ such that the variety $\mathbf{V}(f_{1},\dotsc,f_{Z})$ is $s$ -wise $m$ -independent, is of pure dimension $b-Z$ , and contains at least $\tfrac{1}{2}q^{b-Z}$ many $\mathbb{F}_{q}$ -points.

Proof.

We shall choose each $f_{1},\dotsc,f_{Z}$ uniformly at random from $\mathbb{F}_{q}[x_{0},\dotsc,x_{b}]_{m}$ . Let

	$\displaystyle B$	$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{(f_{1},\dotsc,f_{Z}):\mathbf{V}(f_{1},\dotsc,f_{Z})\text{ is not }s\text{-wise }m\text{-independent}\}$
		$\displaystyle\subseteq\overline{\mathbb{F}_{q}}[x_{0},\dotsc,x_{b}]_{m}^{Z}.$
The set $B$ is a variety, for we can think of it as the image of the projection of the bigger variety
	$\displaystyle B^{\prime}$	$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{(f_{1},\dotsc,f_{Z},p_{1},\dotsc,p_{t}):f_{i}(p_{j})=0\text{ for all }i,j\}$
		$\displaystyle\subseteq\overline{\mathbb{F}_{q}}[x_{0},\dotsc,x_{b}]_{m}^{Z}\times\Phi_{t}(b,m)$

onto the first factor.

Furthermore, recall that $\Phi_{t}(b,m)$ is defined by the linear dependence of the $m$ -th powers of the linear forms associated to the points $p_{i}$ . Since the number of linear forms and the numbers of variables therein do not depend on $q$ , using Bézout’s inequality we may obtain an upper bound on the degree $B^{\prime}$ (and hence on the degree of $B$ ) that is independent of $q$ (but depends on $b,m$ and $Z$ ).

By Lemma 14, $B$ is of codimension at least $1$ in $\overline{\mathbb{F}_{q}}[x_{0},\dotsc,x_{b}]_{m}^{Z}$ . By Lemma 3, a random element $\mathbb{F}_{q}[x_{0},\dotsc,x_{b}]_{m}^{Z}$ is in $B$ with probability $O(\frac{1}{q}\deg B)$ . Since $\deg B$ is independent of $q$ , this probability is $O(1/q)$ .

Similarly, since $\mathbf{V}(f_{1},\dotsc,f_{Z})$ is of codimension $Z$ for generic polynomials $f_{1},\dotsc,f_{Z}$ , it follows that $\mathbf{V}(f_{1},\dotsc,f_{Z})$ is of smaller codimension with probability $O(1/q)$ for random polynomials $f_{1},\dotsc,f_{Z}$ . Furthermore, since $\mathbf{V}(f_{1},\dotsc,f_{Z})$ is defined by $Z$ polynomials, no component of it can have codimension more than $Z$ (see [21, Corollary 1.14]), and so $\mathbf{V}(f_{1},\dotsc,f_{Z})$ is of pure dimension $b-Z$ .

Let $Y=\mathbb{P}^{b}(\mathbb{F}_{q})$ . Applying Item 4(a) we see that

\Pr\bigl{[}\lvert\mathbf{V}(f_{1},\dotsc,f_{Z})\cap\mathbb{P}^{b}(\mathbb{F}_{q})\rvert\leq\tfrac{1}{2}q^{b-Z}\bigr{]}\leq 4q^{Z-b}.

Since $b-Z\geq 1$ , this is also $O(1/q)$ , and so random polynomials $f_{1},\dotsc,f_{Z}$ satisfy the conclusion of the lemma with probability $1-O(1/q)$ . ∎

5. Upper bound on $\phi_{t}(b,m)$

For the purpose of proving an exponential bound in Theorem 1, we need only a bound of the form $\phi_{t}(b,m)\leq(1-\varepsilon)bt$ that is valid for small $t$ . We go a step further: it can be shown that $m+2$ points are $m$ -dependent if and only if they are collinear, and so $\phi_{m+2}(b,m)=2(b-1)+m$ . The bound on $\phi_{t}(b,m)$ that we prove is sufficiently strong that the maximum of the quantity $\frac{1}{t-1}\phi_{t}(b,m)$ in our application is achieved at $t=m+2$ , and so further improvements in the bound would not lead to a smaller base of exponent in Theorem 1.

Definition 16.

We say that points $p_{1},\dotsc,p_{t}\in\mathbb{P}^{b}$ are strongly $m$ -dependent if they span $\mathbb{P}^{b}$ and the associated linear forms $\ell_{i}(x)=\langle x,p_{i}\rangle$ satisfy a relation of the form

(6)

c_{1}\prod_{j=1}^{m}\ell_{1}(x_{j})+\dotsb+c_{t}\prod_{j=1}^{m}\ell_{t}(x_{j})=0

in which all the coefficients $c_{1},\dotsc,c_{t}$ are non-zero.

Note that Lemma 10 implies that every minimal $m$ -dependent set is strongly $m$ -dependent inside whichever space it spans. The key to our upper bound on $\phi_{t}(b,m)$ is a lower bound on the number of points in any strongly $m$ -dependent set. We begin by proving a relatively weak such bound, adapting the argument which is implicit in the work of Białynicki-Birula and Schinzel [3]. We then bootstrap this weak bound to a stronger bound in Lemma 20.

Lemma 17.

For $m\geq 2$ , every strongly $m$ -dependent set in $\mathbb{P}^{b}$ has at least $2(b+1)$ many points.

Proof.

Let $p_{1},\dotsc,p_{t}$ be strongly $m$ -dependent points in $\mathbb{P}^{b}$ . By renumbering the points if necessary, we may assume that the points $p_{1},\dotsc,p_{b+1}$ span $\mathbb{P}^{b}$ . If $t\leq 2b+1$ , then the remaining points $p_{b+2},\dotsc,p_{t}$ do not span $\mathbb{P}^{b}$ because there are only $t-b\leq b$ of them. Hence, there is an $a\in\mathbb{P}^{b}$ such that $\langle a,p_{i}\rangle=0$ for $i=b+2,\dotsc,t$ . Plugging $a$ in for each of $x_{2},x_{3},\dotsc,x_{m}$ , in (6) and writing simply $x$ for $x_{1}$ we obtain a relation of the form

c_{1}^{\prime}\ell_{1}(x)+\dotsb+c_{b+1}^{\prime}\ell_{b+1}(x)=0

where $c_{i}^{\prime}=c_{i}\langle a,p_{i}\rangle^{m-1}$ . Since $p_{1},\dotsc,p_{b+1}$ span $\mathbb{P}^{b}$ , not all the coefficients $c_{i}^{\prime}$ vanish. Because the points $p_{1},\dotsc,p_{b+1}$ are linearly independent, the linear forms $\ell_{1},\dotsc,\ell_{b+1}$ are linearly independent, and so we reach a contradiction with the previously-made assumption that $t\leq 2b+1$ . ∎

To prove the stronger bound, we need to establish a couple of auxiliary results.

Lemma 18.

Let $G$ be an $n$ -vertex graph with at most $n$ edges. Then $G$ contains an independent set of size $n/3$ .

Proof.

According to Turán’s theorem applied to the complement of $G$ , the independence number of $G$ is minimized when $G$ is a union of cliques whose sizes differ by at most $1$ . So, assume that $G$ is of this form. This implies that the largest clique in $G$ is of size at most $3$ , for otherwise $G$ would have had more than $n$ edges. Hence, $G$ is a union of at least $n/3$ disjoint cliques, and so has an independent set of size $\lceil n/3\rceil$ . ∎

Lemma 19.

Suppose that $V$ is a finite-dimensional vector space (over any field), $B$ and $B^{\prime}$ are each a basis for $V$ , and that no vector in $B$ is a multiple of any vector in $B^{\prime}$ , and vice versa. Then there exists a subset $C\subset B$ of size $\lvert C\rvert\geq\tfrac{1}{3}\dim V$ such that $B^{\prime}\cap\operatorname{span}(C)=\emptyset$ .

Proof.

Given a vector $v\in B^{\prime}$ , express it in the basis $B$ as $v=\sum_{b\in B}c_{b,v}b$ , and define the set $S(v)\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{b\in B:c_{b,v}\neq 0\}$ . Equivalently, the set $S(v)$ is the support of $v$ , when expressed in the basis $B$ .

Let $H\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{S(v):v\in B^{\prime}\}$ . Since no vector of $B^{\prime}$ is a multiple of any vector in $B$ , it follows that $\lvert S(v)\rvert\geq 2$ , and so we may view $H$ as a (potentially non-uniform) hypergraph all whose edges have at least two elements. Since $v\in\operatorname{span}(C)$ if and only if $S(v)\subseteq C$ , our task is to find an independent in $H$ of size $n/3$ .

Replace each edge in $H$ by an arbitrary two-element subset thereof, obtaining a graph $G$ . An independent set in $G$ is also an independent set in $H$ . Since $G$ has $\dim V$ vertices and at most $\dim V$ distinct edges, an appeal to Lemma 18 completes the proof. ∎

With this in place, we are ready to prove the stronger bound on $\phi_{t}(b,m)$ .

Lemma 20.

For $m\geq 2$ , every strongly $m$ -dependent set in $\mathbb{P}^{b}$ has at least $\frac{m+4}{3}(b+1)$ many points.

Proof.

The proof is by induction on $m$ . The base case $m=2$ is contained in Lemma 17. So, assume that $m\geq 3$ .

Let $P$ be strongly $m$ -dependent. Let $B\subset P$ be any set of $b+1$ points that span $\mathbb{P}^{b}$ . We break into two cases.

Suppose that the set $P\setminus B$ does not span $\mathbb{P}^{b}$ . In this case we proceed similarly to the proof of Lemma 17. Namely, we choose an $a\in\mathbb{P}^{b}$ such that $\langle a,p\rangle=0$ for all $p\in P\setminus B$ . Plugging $a$ in for each of $x_{2},x_{3},\dotsc,x_{m}$ in (6) and writing $x$ for $x_{1}$ , we obtain a non-trivial linear relation between the $b+1$ linear forms $\langle x,p\rangle$ with $p\in B$ , contradicting the fact that $B$ spans $\mathbb{P}^{b}$ .

Suppose that $P\setminus B$ spans $\mathbb{P}^{b}$ . Let $B^{\prime}$ be any $b+1$ points of $P\setminus B$ that span $\mathbb{P}^{b}$ . Thinking of $B^{\prime}$ as points in a vector space of dimension $b+1$ , from Lemma 19 it follows that there is $C\subset B$ of size $\lvert C\rvert\geq(b+1)/3$ such that $B^{\prime}\cap\operatorname{span}(C)=\emptyset$ . Since the field $\overline{\mathbb{F}_{q}}$ is infinite, there is an $a\in\operatorname{span}(C)^{\bot}$ such that $\langle a,p\rangle\neq 0$ for all $p\in P\setminus\operatorname{span}(C)$ . Setting $x_{m}=a$ in (6) hence yields a relation

\sum_{p\in P\setminus\operatorname{span}(C)}c_{p}^{\prime}\prod_{j=1}^{m-1}\ell_{p}(x_{j})=0,

where $c_{p}^{\prime}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}c_{p}\langle a,p\rangle\neq 0$ . Since the set $P\setminus\operatorname{span}(C)$ contains $B^{\prime}$ , this relation shows that $P\setminus\operatorname{span}(C)$ is a strongly $(m-1)$ -dependent set. Since $\lvert P\rvert\geq\lvert P\setminus\operatorname{span}(C)\rvert+\lvert C\rvert$ , we are done by induction. ∎

Define

	$\displaystyle\Phi^{\prime}_{t}(b,m)$	$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{(p_{1},\dotsc,p_{t})\in\Phi_{t}(b,m):\text{points }p_{1},\dotsc,p_{t}\text{ span }\mathbb{P}^{b}\},$
	$\displaystyle\phi^{\prime}_{t}(b,m)$	$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\dim\Phi^{\prime}_{t}(b,m).$

Lemma 21.

Suppose that $m\geq 3$ . Then $\phi_{t}^{\prime}(b,m)\leq(t-b-1)(b+1)$ .

Proof.

For notational brevity, let $\mathcal{L}$ denote the vector space of all non-zero linear forms in $b+1$ variables. Let $\mathcal{L}_{*}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\mathcal{L}\setminus\{0\}$ and define

	$\displaystyle U$	$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{(\ell_{1},\dotsc,\ell_{t})\in\mathcal{L}_{*}^{t}:\operatorname{span}\{\ell_{1},\dotsc,\ell_{t}\}=\mathcal{L}\},$
	$\displaystyle W_{t}^{\prime}$	$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{(\ell_{1},\dotsc,\ell_{t})\in U:\sum_{i=1}^{t}c_{i}\prod_{j=1}^{m}\ell_{i}(x_{j})=0\text{ for unique }c_{1},\dotsc,c_{t}\neq 0\},$
	$\displaystyle W_{t}$	$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{(\ell_{1},\dotsc,\ell_{t})\in U:\sum_{i=1}^{t}\prod_{j=1}^{m}\ell_{i}(x_{j})=0\}.$

Since each point in $\mathbb{P}^{b}$ corresponds to a $1$ -dimensional family of linear forms (differing up to a multiplication by a non-zero scalar), a collection of $t$ points corresponds to a $t$ -dimensional family of linear forms. Using the characterization of $m$ -dependence in Lemma 10 this implies that $\dim\Phi^{\prime}_{t}(b,m)+t\leq\dim W_{t}^{\prime}$ . Since we also have $\dim W_{t}^{\prime}\leq\dim W_{t}+t$ , it follows that $\dim\Phi^{\prime}_{t}(b,m)\leq\dim W_{t}$ . We shall upper bound $\dim W_{t}$ by bounding the dimension of the tangent space at every point of $W_{t}$ .

Let $(\ell_{1},\dotsc,\ell_{t})\in W_{t}$ . Since $\ell_{1},\dotsc,\ell_{t}$ span $\mathcal{L}$ , by renumbering the forms, we may assume that the forms $\ell_{1},\dotsc,\ell_{b+1}$ span $\mathcal{L}$ . Furthermore, by applying a linear change of coordinates, we may also assume that $\ell_{i}(y_{0},\dotsc,y_{b})=y_{i-1}$ for each $i=1,2,\dotsc,b+1$ . Then $(\Delta_{1},\dotsc,\Delta_{t})\in\mathcal{L}^{t}$ is in the tangent space to $W_{t}$ at the point $(\ell_{1},\dotsc,\ell_{t})\in W_{t}$ if and only if

(7)

\sum_{i=1}^{b+1}\Delta_{i}(x_{k})\prod_{j\neq k}x_{j,i-1}+\sum_{i=b+2}^{t}\sum_{k=1}^{m}\Delta_{i}(x_{k})\prod_{j\neq k}\ell_{i}(x_{j})=0.

Think of this condition as a system of linear equations. Each of the unknowns $\Delta_{1},\dotsc,\Delta_{t}\in\mathcal{L}$ can be thought of as a vector of $b+1$ many scalar unknowns. There is one linear equation for each monomial in the $x$ -variables appearing in (7). For $i\leq b+1$ , all the monomials appearing as coefficients of $\Delta_{i}$ are a product of $m-1$ variables of the form $x_{j,i-1}$ , and a single other variable. Since $m\geq 3$ , this means the monomials appearing as coefficients of $\Delta_{i}$ , for $i\leq b+1$ , are disjoint. This means that the system of equations is of rank at least $(b+1)^{2}$ , i.e., the tangent space to $W_{t}$ at $(\ell_{1},\dotsc,\ell_{t})$ is of codimension at least $(b+1)^{2}$ . Therefore,

\phi^{\prime}_{t}(b,m)\leq\dim W_{t}\leq t(b+1)-(b+1)^{2}=(t-b-1)(b+1).\qed

Lemma 22.

Suppose that $t,b,m\geq 3$ are integers. Then

a)

$\Phi_{t}(b,m)=\emptyset$ if $t\leq m+1$ , and
b)

$\phi_{t}(b,m)\leq\lfloor\tfrac{3}{m+4}t\rfloor(b+1+\tfrac{m-2}{m+4}t)$ if $m+2\leq t\leq b$ .

Proof.

From Lagrange interpolation it follows that no set of $m+1$ or fewer points is $m$ -dependent. Hence $\Phi_{t}(b,m)=\emptyset$ if $t\leq m+1$ .

Let $r^{\prime}=\lfloor\tfrac{3}{m+4}t-1\rfloor$ . By Lemma 20, every minimal $m$ -dependent set of size $t$ spans a subspace of projective dimension at most $r^{\prime}$ . Let $\operatorname{Gr}(r,\mathbb{P}^{b})$ be the Grassmanian of linear subspaces of dimension $r$ in $\mathbb{P}^{b}$ . We then have

	$\displaystyle\phi_{t}(b,m)$	$\displaystyle=\dim\Phi_{t}(b,m)$
		$\displaystyle=\dim\bigcup_{r\leq r^{\prime}}\{(p_{1},\dotsc,p_{t},H)\in\Phi_{t}(b,m)\times\operatorname{Gr}(r,\mathbb{P}^{b}):p_{1},\dotsc,p_{t}\text{ span }H\},$
		$\displaystyle=\max_{r\leq r^{\prime}}\bigl{(}\phi^{\prime}_{t}(r,m)+\dim\operatorname{Gr}(r,\mathbb{P}^{b})\bigr{)}$
		$\displaystyle\leq\max_{r\leq r^{\prime}}\bigl{(}(t-r-1)(r+1)+(r+1)(b-r)\bigr{)}\qquad\qquad\text{by \lx@cref{creftypecap~refnum}{lem:vdtprime}}$
		$\displaystyle=\max_{r\leq r^{\prime}}(t+b-2r-1)(r+1).$
Without the restriction on $r$ , the maximum of the quadratic $(t+b-2r-1)(r+1)$ is achieved when $r=\frac{t+b-3}{4}$ . Since $b\geq t$ , the value $\frac{t+b-3}{4}$ is larger than $r^{\prime}$ , and so
	$\displaystyle\phi_{t}(b,m)$	$\displaystyle\leq(t+b-2r^{\prime}-1)(r^{\prime}+1)\leq(t+b+1-\tfrac{6}{m+4}t)\lfloor\tfrac{3}{m+4}t\rfloor.\hskip-60.00009pt$

6. Construction for the Turán problem

Lemma 23.

Let $r,s,Z\geq 1$ be integers. Set $b=r+s+Z$ . Suppose that $b,m,Z,r$ satisfy the inequalities $\binom{m+1+r}{m}\geq s^{2}$ , $m\geq 3$ as well as the condition (5) for all fields of sufficiently large characteristic. Then, for every sufficiently large $n$ there is a graph with $\Theta(n)$ vertices and $\Omega(n^{2-1/s})$ edges that is $K_{s,t}$ -free for every $t>m^{s+Z}\prod_{i=1}^{r}M_{i}(s^{2})$ .

Proof.

Using Bertrand’s postulate, pick a prime $q$ such that $n\leq q^{s}<2^{s}n$ . Since $n$ is sufficiently large, we may assume that the condition (5) is satisfied for $\overline{\mathbb{F}_{q}}$ .

Let $f_{1},\dotsc,f_{Z}$ be polynomials as in Lemma 15. Let $\delta_{i}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}M_{r-i+1}(s^{2})$ for $i=1,2,\dotsc,r$ . Let $h_{1},\dotsc,h_{r}$ and $h_{1}^{\prime},\dotsc,h_{r}^{\prime}$ be two independent collections of random homogeneous polynomials on $\mathbb{P}^{b}$ with $\mathbb{F}_{q}$ -coefficients of degrees $\deg h_{i}=\deg h_{i}^{\prime}=\delta_{i}$ . Let $g$ be a random bihomogeneous $\mathbb{F}_{q}$ -polynomial on $\mathbb{P}^{b}\times\mathbb{P}^{b}$ of bidegree $(m,m)$ .

Define

	$\displaystyle L_{0}$	$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\mathbf{V}(f_{1},\dotsc,f_{Z},h_{1},\dotsc,h_{r})\cap\mathbb{P}^{b}(\mathbb{F}_{q}),$
	$\displaystyle R_{0}$	$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\mathbf{V}(f_{1},\dotsc,f_{Z},h_{1}^{\prime},\dotsc,h_{r}^{\prime})\cap\mathbb{P}^{b}(\mathbb{F}_{q}).$

Let $c^{\prime}=c^{\prime}(m,s,\deg W)$ be the constant from Lemma 13 applied with $W=\mathbf{V}(f_{1},\dotsc,f_{Z})$ , and let $C=C(s^{2},s,m^{s+Z},\delta_{1},\dotsc,\delta_{r})$ be the constant from Item 12(b). Put $c\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\min\bigl{(}\tfrac{1}{4},c^{\prime},(5C)^{-1/s}\bigr{)}$ . Let $L$ be a set of size $\min(cq^{s},\lvert L_{0}\rvert)$ chosen canonically from $L_{0}$ (for example, it could be the initial segment of $L_{0}$ in some fixed ordering of $\mathbb{P}^{b}$ ). Let $R$ be a similarly defined subset of $R_{0}$ of size $\min(cq^{s},\lvert R_{0}\rvert)$ .

Define the bipartite graph $G$ with parts $L$ and $R$ by connecting the pair $(l,r)\in L\times R$ whenever $g(l,r)=0$ .

$G$ has $\Theta(q^{s})$ vertices and $\Omega(q^{2s-1})$ edges.

By Item 4(a), both $L_{0}$ and $R_{0}$ have at least $\tfrac{1}{4}q^{s}$ elements with probability $1-O(1/q^{s})$ . Since $c\leq\tfrac{1}{4}$ , this means that

(8)

\Pr\bigl{[}\lvert L\rvert=\lvert R\rvert=cq^{s}\bigr{]}\geq 1-O(q^{-s}).

Since the polynomial $g$ is independent of $L_{0}$ and $R_{0}$ , it follows, by Item 4(b) applied with $Y=L\times R$ to the single random polynomial $g$ , that the edge set $E(G)=(L\times R)\cap\mathbf{V}(g)$ is of size at least $\lvert L\rvert\lvert R\rvert/2q$ with probability at least $1-O(q^{-2s+1})$ . In view of (8), it then follows that

(9)

\Pr[E(G)\geq\tfrac{1}{2}c^{2}q^{2s-1}]\geq 1-O(q^{-s}).

$G$ is $K_{s,t}$ -free.

Because of the symmetry between the two parts of $G$ , it suffices to show that $G$ is very unlikely to contain $K_{s,t}$ with the part of size $s$ embedded into $L$ . Since $L\subset\mathbf{V}(f_{1},\dotsc,f_{Z})$ and $\mathbf{V}(f_{1},\dotsc,f_{Z})$ is $m$ -independent, the set $L$ is $m$ -independent. Let $W\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\mathbf{V}(f_{1},\dotsc,f_{Z})$ . Since $\binom{m+1+r}{m}\geq s^{2}$ , Lemma 13 applies, telling us that with probability $\tfrac{4}{5}$ every variety of the form

W_{l_{1},\dotsc,l_{s}}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{w\in W:g(l_{1},w)=\dotsb=g(l_{s},w)=0\},

for distinct $l_{1},\dotsc,l_{s}$ , is of dimension $\dim W-s=r$ . Let $\mathcal{W}$ be the set of all varieties of the form $W_{l_{1},\dotsc,l_{s}}$ for distinct $l_{1},\dotsc,l_{s}\in L$ . The set $\mathcal{W}$ is random: it depends on the random choice of polynomials $h_{1},\dotsc,h_{r}$ (because $L$ depends on these polynomials), and it depends on the polynomial $g$ . Crucially, $\mathcal{W}$ does not depend on the polynomials $h_{1}^{\prime},\dotsc,h_{r}^{\prime}$ .

So, we may apply Item 12(b) to each variety in $\mathcal{W}$ and polynomials $h_{1}^{\prime},\dotsc,h_{r}^{\prime}$ . Note that $\deg(W_{l_{1},\dotsc,l_{s}})\leq\deg(W)m^{s}\leq m^{s+Z}$ by Bézout’s inequality. Therefore, combined with the union bound, the proposition tells us that

	$\displaystyle\Pr[\exists$	$\displaystyle\text{ distinct }l_{1},\dotsc,l_{s}\in L\text{ s.t.\ \!}\dim\bigl{(}W_{l_{1},\dotsc,l_{s}}\cap\mathbf{V}(h_{1}^{\prime},\dotsc,h_{r}^{\prime})\bigr{)}>0]$
		$\displaystyle\leq\Pr[\dim(W_{l_{1},\dotsc,l_{s}})\neq r\text{ for some }l_{1},\dotsc,l_{s}]$
		$\displaystyle\qquad+\Pr\bigl{[}\lvert L\rvert\neq cq^{s}\bigr{]}+(cq^{s})^{s}\cdot C(s^{2},s,m^{s+Z})q^{-s^{2}}$
		$\displaystyle\leq\tfrac{1}{5}+O(q^{-s})+c^{s}C(s^{2},s,m^{s+Z}).$

Because the constant $c$ satisfies $c\leq(5C)^{-1/s}$ , this probability is at most $\tfrac{2}{5}+O(q^{-s})$ . Note that the variety $W_{l_{1},\dotsc,l_{s}}\cap\nobreak\mathbf{V}(h_{1}^{\prime},\dotsc,h_{r}^{\prime})$ contains all common neighbors of the vertices $l_{1},\dotsc,l_{s}$ in the graph $G$ . By Bézout’s inequality, the degree of this variety is at most $\deg(W)m^{s}\prod_{i=1}^{r}\delta_{i}<t$ , and hence

\Pr[\text{some }s\text{ vertices in }L\text{ have }t\text{ common neighbors in }G]\leq\tfrac{2}{5}+O(q^{-s}).

By symmetry, we may derive the same bound with the roles of $L$ and $R$ reversed, and so

(10)

\Pr[G\text{ contains }K_{s,t}]\leq\tfrac{4}{5}+O(q^{-s}).

Putting (8), (9) and (10) together, it follows that graph $G$ has $\Theta(q^{s})=\Theta(n)$ vertices, at least $\Omega(q^{2s-1})=\Omega(n^{2-1/s})$ edges, and contains no $K_{s,t}$ with probability at least $\tfrac{1}{5}-O(q^{-s})>0$ . In particular, such a graph $G$ exists. ∎

Lemma 24.

For every $r\geq 1$ and every $T$ , we have $\prod_{k=1}^{r}M_{k}(T)\leq T^{1+\log r}r!$ .

Proof.

Since $\binom{m+k}{m}\geq\bigl{(}\frac{m+1}{k}\bigr{)}^{k}$ , the function $M_{k}$ satisfies $M_{k}(T)\leq\lfloor kT^{1/k}\rfloor.$ The lemma then follows from the inequality $1+\tfrac{1}{2}+\dotsb+\tfrac{1}{r}\leq 1+\log r$ . ∎

Proof of Theorem 1.

We may assume that $s\geq 100$ , for otherwise the theorem follows from the result of Alon, Rónyai and Szabó [2] that we mentioned in the introduction. Let $m\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}3$ , $r\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\lfloor(6s^{2})^{1/3}\rfloor$ , and $Z\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}s+r+3$ .

Observe that

\tfrac{1}{t-1}\lfloor\tfrac{3}{7}t\rfloor\leq\begin{cases}\tfrac{1}{2}&\text{ for }t=5,6,7,\\ \tfrac{7}{15}&\text{ for }t=8,9,10,11,\dotsc.\\ \end{cases}

Hence, for $5\leq t\leq 7$ , Item 22(b) implies that

	$\displaystyle\tfrac{1}{t-1}\phi_{t}(b,m)$	$\displaystyle\leq\tfrac{1}{2}(s+r+Z+1+\tfrac{1}{7}t)\leq\tfrac{1}{2}(s+r+2)+\tfrac{1}{2}Z<Z.$
Similarly, for $8\leq t\leq s$ we have
	$\displaystyle\tfrac{1}{t-1}\phi_{t}(b,m)$	$\displaystyle\leq\tfrac{7}{15}(s+r+Z+1+\tfrac{1}{7}t)\leq\tfrac{7}{15}(\tfrac{8}{7}s+r+1)+\tfrac{7}{15}Z$
		$\displaystyle<\tfrac{8}{15}(s+r+2)+\tfrac{7}{15}Z<Z.$

This shows that the condition (5) holds for $t=5,6,\dotsc,s$ . It also holds for $t=2,3,4$ by Item 22(a).

We have $\binom{m+1+r}{m}=\binom{r+4}{3}\geq(r+1)^{3}/6\geq s^{2}$ , and so the result follows from Lemmas 23 and 24, the estimation

r!(s^{2})^{1+\log r}\leq 2s^{1/2}(r/e)^{r}\cdot(s^{2})^{1+\log r}\leq s^{\tfrac{2}{3}r+2+2\log r}\leq s^{r}\qquad\text{for }s\geq 23,

and the inequality $3^{r+3}s^{r}\leq s^{3r/2}\leq s^{4s^{2/3}}$ that is valid for $s\geq 100$ . ∎

7. Construction for the Zarankiewicz problem

This is similar, but simpler than the construction for the Turán problem from the preceding section because we do not need Lemmas 15 and 22.

Lemma 25.

Let $r,s,T\geq 1$ be integers satisfying $T\leq\binom{r+1+m}{m}$ . Then, for every sufficiently large $n$ there is a sided graph with $\Theta(n^{T/s^{2}})$ vertices on the left, $\Theta(n)$ vertices on the right, and $\Omega(n^{T/s^{2}+1-1/s})$ edges that contains no sided $K_{s,t}$ for every $t>m^{s}\prod_{i=1}^{r}M_{i}(T)$ .

Proof.

Let $q$ be the power of $2$ satisfying $n\leq q^{s}<2^{s}n$ . Let $b\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}r+s$ .

Set $\delta_{i}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}M_{r-i+1}(T)$ for $i=1,\dotsc,r$ . Let $c^{\prime}=c^{\prime}(m,s,1)$ be the constant from Lemma 13, and let $C=C\bigl{(}T,s,m^{s},\delta_{1},\dotsc,\delta_{r}\bigr{)}$ be the constant from Item 12(b). Put $c\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\min(c^{\prime},(5C)^{-1/s})$ . Pick $a\geq 1$ and an $s$ -wise $m$ -independent set $L$ in $\mathbb{P}^{a}(\mathbb{F}_{q})$ of size $cq^{T/s}$ arbitrarily. For example, we may choose $a=cq^{T/s}$ and let $L$ consist of the standard basis vectors.

We next pick several random polynomials with coefficients in $\mathbb{F}_{q}$ : Let $g$ be a random bihomogeneous polynomial on $\mathbb{P}^{a}\times\mathbb{P}^{b}$ of bidegree $(m,m)$ , and let $h_{1}^{\prime},\dotsc,h_{r}^{\prime}$ be random independently-chosen homogeneous polynomials on $\mathbb{P}^{b}$ of degrees $\deg h_{i}^{\prime}=\delta_{i}$ . Let

R\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\mathbf{V}(h_{1}^{\prime},\dotsc,h_{r}^{\prime})\cap\mathbb{P}^{b}(\mathbb{F}_{q}).

Define the sided graph $G$ with the left part $L$ and the right part $R$ by connecting $(l,r)\in L\times R$ whenever $g(l,r)=0$ .

Invoking Lemma 13 with $W=\mathbb{P}^{b}$ we see that, with probability $\tfrac{4}{5}$ , every variety of the form

W_{l_{1},\dotsc,l_{s}}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\{w\in\mathbb{P}^{b}:g(l_{1},w)=\dotsb=g(l_{s},w)=0\},

for distinct $l_{1},\dotsc,l_{s}\in L$ , has dimension $b-s=r$ . By Bézout’s inequality, $\deg W_{l_{1},\dotsc,l_{s}}\leq m^{s}$ , and so the union bound and Item 12(b) together imply that

(11)

\Pr[G\text{ contains a sided }K_{s,t}]\leq\tfrac{2}{5}+O(q^{-s}),

where we used that $c\leq(5C)^{-1/s}$ , similarly to the corresponding step in the proof of Lemma 23.

Also, from Item 12(b) applied to $W=\mathbb{P}^{b}$ we obtain the inequality $\Pr[\dim\mathbf{V}(h_{1}^{\prime},\dotsc,h_{r}^{\prime})>s]=\nobreak O(q^{-s})$ . In view of Lemma 3, this implies that

(12)

\Pr\bigl{[}\lvert R\rvert=O_{s}(q^{s})\bigr{]}\geq 1-O(q^{-s}).

Finally, by applying Items 4(a) and 4(b), it follows that

	$\displaystyle\Pr\bigl{[}\lvert E(G)\rvert\!\geq\!\tfrac{1}{4}cq^{T/s+s-1}\bigr{]}$	$\displaystyle\mkern-2.1mu\geq\mkern-2.1mu\Pr\bigr{[}\lvert R\rvert\!\geq\!\tfrac{1}{2}q^{s}\bigr{]}\!\Pr\bigl{[}\lvert E(G)\rvert\!\geq\!\tfrac{1}{2}cq^{T/s}\!\cdot\!\tfrac{1}{2}q^{s-1}\!\bigm{\|}\!\lvert R\rvert\!\geq\!\tfrac{1}{2}q^{s}\bigr{]}$
		$\displaystyle\mkern-2.1mu\geq\mkern-2.1mu\bigl{(}1-O(q^{-s})\bigr{)}\bigl{(}1-O(q^{-s+1}q^{-T/s})\bigr{)}$
(13)			$\displaystyle\mkern-2.1mu=\mkern-2.1mu1-O(q^{-s+1}).$

From (11), (12), (13) we see that there exists a sided graph with $cq^{T/s}$ vertices on the left, $O_{s}(q^{s})$ vertices on the right, and at least $\tfrac{1}{4}cq^{T/s+s-1}$ edges. ∎

Proof of Item 2(a).

Let $T\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}s^{2}\log_{n}m$ , and $r\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\lfloor s/\log^{2}s\rfloor$ . Note that these constants satisfy $\binom{r+1+k}{r}\geq(s/\log^{2}s)^{k}/k!\geq s^{2}\log_{n}m$ . We then use Lemma 25 with $k$ in place of $m$ , and appeal to Lemma 24 to bound

	$\displaystyle\prod_{i=1}^{r}M_{i}(T)$	$\displaystyle\leq T^{1+\log r}r!\leq s^{(1+\log r)k}r^{r}$
		$\displaystyle\leq s^{(1+\log s)\frac{s}{2\log^{3}s}}(s/\log^{2}s)^{s/\log^{2}s}\leq e^{2s/\log s}$

to obtain the stated result. ∎

Proof of Item 2(b).

Let $k\geq 1$ be an integer to be chosen shortly. Let $r\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\lceil s/\log s\rceil$ , and $T\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\binom{r+1+k}{k}$ Note that there are constants $c_{s}^{\prime\prime}$ and $c_{s}^{\prime\prime\prime}$ such that

c_{s}^{\prime\prime}k^{r}\leq T/s^{2}\leq c_{s}^{\prime\prime\prime}k^{r},

and choose $k$ to be the smallest integer so that $c_{s}^{\prime\prime}k^{r}\geq\log_{n}m$ .

Applying Lemma 25 with $k$ in place of $m$ , we obtain a sided graph whose left part is of size $\Omega(n^{T/s^{2}})=\nobreak\Omega(n^{c_{s}^{\prime\prime}k^{r}})=\Omega(m)$ and the right part is of size $\Theta(n)$ , matching the Kövári, Sós, Turán bound for $K_{s,t}$ -free graphs for $t>k^{s}\prod_{i=1}^{r}M_{i}(r)$ . Since

k^{s}\prod_{i=1}^{r}M_{i}(r)\leq k^{s}r^{r}T^{1+\log r}=O_{s}(k^{r+2r\log s})=O_{s}(\log_{n}m)^{1+2\log s},

this completes the proof. ∎

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Extremal graphs without exponentially-small bicliques

Abstract.

2020 Mathematics Subject Classification:

1. Introduction

The Turán problem.

Theorem 1.

The Zarankiewicz problem.

Theorem 2.

Main proof idea, in a nutshell.

Proof ideas, in more detail.

Paper organization.

Acknowledgments.

2. Algebraic tools

Varieties and their 𝔽q\mathbb{F}_{q}-points.

Lemma 3 (Weakening of [10, Corollary 3.3]).

Lemma 4.

Proof.

Bézout’s inequality.

Lemma 5 (Bézout’s inequality, [13, p. 228, Example 12.3.1]).

Hilbert functions.

Lemma 6.

Proof.

Lemma 7.

Proof.

3. Definition and uses of mm-independence

Definition 8.

Definition 9.

Lemma 10.

Proof of (a)⇔\iff(b)..

Proof of (b)⟹\implies(c)..

Proof of (c)⟹\implies(b)..

Proposition 11.

Proof.

Proposition 12.

Proof.

Lemma 13.

Proof.

4. Construction of mm-independent varieties

Lemma 14.

Proof of Lemma 14.

Lemma 15.

Proof.

5. Upper bound on ϕt​(b,m)\phi_{t}(b,m)

Definition 16.

Lemma 17.

Proof.

Lemma 18.

Proof.

Lemma 19.

Proof.

Lemma 20.

Proof.

Lemma 21.

Proof.

Lemma 22.

Proof.

6. Construction for the Turán problem

Lemma 23.

Proof.

GG has Θ​(qs)\Theta(q^{s}) vertices and Ω​(q2​s−1)\Omega(q^{2s-1}) edges.

GG is Ks,tK_{s,t}-free.

Lemma 24.

Proof.

Proof of Theorem 1.

7. Construction for the Zarankiewicz problem

Lemma 25.

Proof.

Proof of Item 2(a).

Proof of Item 2(b).

References

Varieties and their $\mathbb{F}_{q}$ -points.

3. Definition and uses of $m$ -independence

Proof of (a) $\iff$ (b)..

Proof of (b) $\implies$ (c)..

Proof of (c) $\implies$ (b)..

4. Construction of $m$ -independent varieties

5. Upper bound on $\phi_{t}(b,m)$

$G$ has $\Theta(q^{s})$ vertices and $\Omega(q^{2s-1})$ edges.

$G$ is $K_{s,t}$ -free.