# Ramsey numbers and extremal structures in polar spaces

John Bamberg1 1Department of Mathematics & Statistics, The University of
Western Australia, Perth, Australia.  ,  Anurag Bishnoi2 2Delft Institute of
Applied Mathematics, Delft University of Technology, Netherlands.  and
Ferdinand Ihringer3 3Department of Mathematics, Southern University of Science
and Technology, Shenzhen, China<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

We use $p$-rank bounds on partial ovoids and the classical bounds on Ramsey
numbers to obtain various upper bounds on partial $m$-ovoids in finite polar
spaces. These bounds imply non-existence of $m$-ovoids for various new
families of polar spaces. We give a probabilistic construction of large
partial $m$-ovoids when $m$ grows linearly with the rank of the polar space.

In the special case of the symplectic spaces over the binary field, we show an
equivalence between partial $m$-ovoids and a generalisation of the Oddtown
theorem from extremal set theory that has been studied under the name of
nearly $m$-orthogonal sets over finite fields. We give new constructions for
partial $m$-ovoids in these spaces and thus $m$-nearly orthogonal sets, for
small values of $m$. These constructions use triangle-free graphs whose
complements have low $\mathbb{F}_{2}$-rank and we give an asymptotic
improvement over the state of the art. We also prove new lower bounds in the
recently introduced rank-Ramsey problem for triangles vs cliques.

##  1\. Introduction

One of the main areas of research in finite geometry is to understand various
extremal substructures inside a given finite geometrical space. For example,
understanding the largest possible sets of points in finite projective or
affine spaces with the property that no three points are collinear is a
classical problem with connections to additive combinatorics [2, 19] and
coding theory [24]. We will study problems of a similar flavor in the
geometrical spaces known as the finite classical polar spaces, which were
introduced by J. Tits [32] as natural geometries corresponding to the finite
classical simple groups. Their main geometric property, as shown by Buekenhout
and Shult [11], is the so-called _one or all axiom_ : if $P$ is a point and
$\ell$ a line not incident with $P$, then $P$ is collinear with either all
points of $\ell$ or with exactly one point of $\ell$. In fact, abstract polar
spaces are defined as point-line geometries satisfying this axiom. We will be
interested in finite classical polar spaces that arise from equipping a finite
vector space with a sesquilinear or quadratic form. The points of the polar
space are then the one-dimensional totally isotropic or singular vector
subspaces and the lines are the two-dimensional totally isotropic or totally
singular vector subspaces, with respect to the given form.

Besides applications to group theory, polar spaces give rise to various
interesting combinatorial objects like strongly regular graphs [10], two-
weight codes [12], and Turán graphs [4]. The incidence structures known as
generalised quadrangles are special cases of polar spaces, and since their
inception, they have been used to construct various extremal objects in graph
theory. Recently, other polar spaces have also found combinatorial
application, with the symplectic space over the binary field being implicitly
used in the work of Conlon and Ferber on multicolour Ramsey numbers [15], and
graphs related to the rank one Hermitian spaces being used in the breakthrough
work of Mattheus and Verstraete on off-diagonal Ramsey numbers [26].

In this paper, we will show applications in the reverse direction, that is,
applying bounds and constructions from extremal combinatorics to improve our
knowledge of substructures of finite classical polar spaces known as
$m$-ovoids. One of our main contribution is a novel constructions of partial
$m$-ovoids that uses triangle-free graphs for which the adjacency matrix of
the complement of the graph has low rank over the binary field.

The study of ovoids in polar spaces dates back to the work of Tits on
geometric constructions of certain Suzuki groups [31, 33, 34], by using a
particular polar space. In [29], Thas synthesised these objects as ovoids of
polar spaces: a set $\mathcal{O}$ of points such that every maximal totally
isotropic subspace meets $\mathcal{O}$ in exactly one point. In terms of the
collinearity graph of the polar space 111The collinearity graph of any point-
line geometry is the graph on points with two points adjacent if they lie on a
common line., these are independent sets that meet every maximal clique in
exactly one vertex, and in particular, they are the largest possible
independent sets. The study of $m$-ovoids begins implicitly with the work of
Segre [27], but was introduced for finite generalised quadrangles by Thas
[30], and further extended by Shult and Thas [28] to all finite polar spaces.
Here, an $m$-ovoid is defined to be a set $\mathcal{O}$ of points such that
every maximal totally isotropic subspace meets $\mathcal{O}$ in exactly $m$
points. This was also extended to strongly regular graphs by Eisfeld [18] in
1998. It is one of the purposes of this paper to give bounds on the parameter
$m$ for which $m$-ovoids can exist by using simple bounds from Ramsey theory.
We emphasise that we are interested in the asymptotic growth of $m$ in terms
of, say, the rank of the polar space or the characteristic of the defining
field. Bamberg, Kelly, Law, and Penttila [6] gave the following lower bound on
$m$ for which an $m$-ovoid can exist in elliptic quadrics, symplectic spaces
(in even dimension), and Hermitian spaces (in odd dimension):

$m\geq\frac{\sqrt{4q^{e+r-1}+9}-3}{2(q-1)}.$

Note, $q=p^{h}$ is the size of the defining field, where $p$ is a prime, and
$r$ is the rank of the polar space. The parameter $e\in\\{1,\frac{3}{2},2\\}$
depends on the type of the polar space (see Section 2). Asymptotically, we
have $m=\Omega(q^{r/2})$ if we fix $q$ and vary $r$, and
$m=\Omega\left(p^{h(e+r-3)/2}\right)$ if we fix $(r,h)$ and vary $p$. This was
recently improved by De Beule, Mannaert, and Smaldore [16]:
$m=\Omega(\sqrt{r}q^{r/2})$ if we fix $q$ and vary $r$, and no change in the
lower bound on $m$ (asymptotically) if we fix $(r,h)$ and vary $p$. While all
of these results give necessary conditions for the existence of $m$-ovoids,
they do not give any upper bounds on the maximum size of a partial $m$-ovoid,
which is a natural problem to study for polar spaces that have no $m$-ovoids.
By using the classical upper bounds on Ramsey numbers, we note in Theorem 3.1,
that a _partial $m$-ovoid_ $\mathcal{O}_{m}$ satisfies

$|\mathcal{O}_{m}|\leq R\left(m+1,(n+p)^{h(p-1)}\right)<(n+p)^{hm(p-1)}.$

From this simple bound we can deduce that (Corollary 3.2), for an $m$-ovoid of
a finite polar space of rank $r$ to exist, we must have
$m+2\log_{p}(2m)\geq\frac{r-1}{p-1}$. While our bound on $m$ is much weaker
than the earlier bounds, it has the advantage of being applicable to all
finite classical polar spaces. Therefore, for all the cases not-covered by the
previous results, where the earlier techniques cannot show any non-existence
results, we get new non-existence results. In Section 4 we give other upper
bounds on partial $2$-ovoids in polar spaces using a combination of spectral
methods and the rank bound on $1$-ovoids, which gives a new simple proof of
known non-existence results on $2$-ovoids [6].

In Section 5.1, we give a general probabilistic construction of partial
$m$-ovoids, which shows that for $m\geq cr$, with any constant $c>1/2$, there
is a partial $m$-ovoid of size $q^{\Omega(r)}$ in a polar space of rank $r$.
We then focus on a special case for constructions. Suppose $q=2$, we are in a
symplectic space $\operatorname{\mathsf{W}}(2r-1,2)$ defined over the vector
space $\mathbb{F}_{2}^{2r}$. For $m=2$, the Ramsey number $R(3,2r+2)$ yields
the bound $|\mathcal{O}_{2}|\leq O(r^{2}/\log r)$. In Section 5.2, we give an
explicit construction of partial $2$-ovoids in $W(2r-1,2)$ of size at least
$(2r)^{1.12895}$. We also show that for all $\varepsilon>0$, there exists an
$m>1$ such that for all large enough $r$, $\operatorname{\mathsf{W}}(2r-1,2)$
possesses a partial $m$-ovoid of size at least
$(2r)^{\frac{\log(16)}{\log(10)}-\varepsilon}$. The convergence is quite sharp
and it leads to new constructions for various small values of $m$. The
construction is somewhat surprising to a finite geometer: a triangle-free
graph with low complementary $2$-rank is constructed, and then infinitely many
graphs with low complementary $2$-rank are produced via the strong product
(through a result of [7]). These graphs can be embedded in binary symplectic
spaces of a certain dimension, because of their connection to _nearly
orthogonal sets_ [14]. We have discovered a new triangle-free graph with low
complementary rank which also improves the known bounds on nearly orthogonal
sets and the recently studied rank-Ramsey problem for triangles [7]. In this
special case of $\operatorname{\mathsf{W}}(2r-1,2)$ partial $m$-ovoids are
also linked to a certain generalisation of the the well-known Oddtown set
families [8]. In Section 5.2, we state this generalised version of the Oddtown
problem and establish its connection to partial $m$-ovoids.

##  2\. Background on finite classical polar spaces

Throughout, we will use the symbol $q$ for a prime power $q:=p^{h}$, $p$ prime
and $h\geq 1$, and we will denote the finite field of order $q$ as
${\mathbb{F}}_{q}$. The vector space of dimension $d$ over ${\mathbb{F}}_{q}$
will be written as $V(n,q)$, and $\operatorname{\mathrm{PG}}(n,q)$ will denote
the projective space with underlying vector space $V(n+1,q)$. Let $f$ be a
(reflexive) sesquilinear or quadratic form on $V(n+1,q)$. The elements of the
finite classical polar space $\operatorname{\mathcal{P}}$ associated with $f$
are the totally singular or totally isotropic subspaces of
$\operatorname{\mathrm{PG}}(n,q)$ with relation to $f$, according to whether
$f$ is a quadratic or sesquilinear form. The Witt index of the form $f$
determines the vector space dimension of the maximal subspaces contained in
$\operatorname{\mathcal{P}}$; the rank $\operatorname{\mathcal{P}}$ equals the
Witt index of its form, and the (projective) dimension of generators will be
one less than the Witt index. Hence, a finite classical polar space of rank
$r$ embedded in $\operatorname{\mathrm{PG}}(n,q)$ has an underlying form of
Witt index $r$, and contains points, lines, …, $(r-1)$-dimensional subspaces.
The elements of maximal dimension are called its _generators_. Recall that a
partial $m$-ovoid is a set of points that meets every generator in at most $m$
points, and it is called an $m$-ovoid if it meets every generator in exactly
$m$ points. Also recall that for $m=1$ these are simply called partial ovoids,
and ovoids, respectively.

We will use projective notation for finite polar spaces so that they differ
from the standard notation for their collineation groups. For example, we will
use the notation $\operatorname{\mathsf{W}}(n-1,q)$ to denote the symplectic
polar space coming from the vector space $V(n,q)$ equipped with a non-
degenerate alternating form. Here is a summary of the notation we will use for
finite polar spaces, together with their ranks and ovoid numbers, that is, the
size of a putative ovoid in the polar space. In this table the parameter $e$
helps us compute various other combinatorial parameters of the polar space,
and essentially arises from knowing the number of points of the ‘rank 1’ polar
space of the same type (which is $q^{e}+1$, where $q$ is the order of the
defining field). For example, a polar space of rank $r$ and type $e$ over a
finite field $\mathbb{F}_{q}$ has $(q^{r}-1)(q^{r+e-1}-1)/(q-1)$ points and
$\prod_{i=1}^{r}(q^{i+e-1}+1)$ generators. The size of a putative ovoid,
written as the Ovoid Number, is equal to $q^{r+e-1}-1$.

Table 1. Notation for the finite classical polar spaces, together with their ovoid numbers. Polar Space | Notation | Ovoid Number | Type $e$ | Rank $r$
---|---|---|---|---
Symplectic |  $\operatorname{\mathsf{W}}(n-1,q)$, $n$ even | $q^{n/2}+1$ | $1$ | $n/2$
Hermitian |  $\operatorname{\mathsf{H}}(n-1,q^{2})$, $n$ odd | $q^{n}+1$ | $3/2$ | $(n-1)/2$
Hermitian |  $\operatorname{\mathsf{H}}(n-1,q^{2})$, $n$ even | $q^{n-1}+1$ | $1/2$ | $n/2$
Orthogonal, elliptic |  $\mathsf{Q}^{-}(n-1,q)$, $n$ even | $q^{n/2}+1$ | $2$ | $(n-2)/2$
Orthogonal, parabolic |  $\mathsf{Q}(n-1,q)$, $n$ odd | $q^{(n-1)/2}+1$ | $1$ | $(n-1)/2$
Orthogonal, hyperbolic |  $\mathsf{Q}^{+}(n-1,q)$, $n$ even | $q^{n/2-1}+1$ | $0$ | $n/2$

Let $\operatorname{\mathcal{P}}$ be a finite polar space defined by a
sesquilinear or quadratic form $f$, and let $X$ be a point of the ambient
projective space. Then $X^{\perp}$ is the set of projective points whose
coordinates are orthogonal to $X$ with respect to the form $f$. Note that when
$f$ is a quadratic form, it determines a (possibly degenerate when $q$ is
even) bilinear form222When $f$ is a quadratic form, $g(v,w):=f(v+w)-f(v)-f(w)$
is a bilinear form, symmetric when $q$ is odd, alternating when $q$ is even.
$g$, and two projective points $X$ and $Y$ are orthogonal with relation to $f$
if, by definition, they are orthogonal with relation to $g$. A partial ovoid
is thus a set of pairwise non-orthogonal points of a polar space. We refer to
[25] for further details on polar spaces.

###  2.1. Some $p$-rank bounds for partial ovoids of finite polar spaces

Our results in the next two sections rely on $p$-rank bounds for partial
ovoids which are summarised in this section, cf. [3, 9].

######  Theorem 2.1 (Folklore, cf. [3, 9]).

Let $\mathcal{O}$ be a partial ovoid in a symplectic space in a vector space
of even dimension $n\geq 4$ over a field with $q=p^{h}$, where $p$ is prime,
elements. Then

$\displaystyle|\mathcal{O}|\leq\binom{p+n-1}{p-1}^{h}+1.$

######  Theorem 2.2 ([3, Theorem 1.6]).

Let $\mathcal{O}$ be a partial ovoid in a quadric in a vector space of
dimension $n\geq 4$ over a field with $q=p^{h}$, where $p$ is prime, elements.
Then

  1. (a)

If $p=2$ and $n$ even, then $|\mathcal{O}|\leq n^{h}+1$.

  2. (b)

If $p=2$ and $n$ odd, then $|\mathcal{O}|\leq(n-1)^{h}+1$.

  3. (c)

If $p>2$, then

$|\mathcal{O}|\leq\left[\binom{n+p-2}{p-1}-\binom{n+p-4}{p-3}\right]^{h}+1.$

  4. (d)

If $p>2$ and there exists a positive integer $u$ satisfying

$u+1\equiv n\pmod{2}\text{ and }n-3\leq up\leq n+p-5,$

then

$|\mathcal{O}|\leq\left[\binom{n+p-2}{p-1}-\binom{n+p-4}{p-3}-\binom{up+2}{n-1}+\binom{up}{n-1}\right]^{h}+1.$

######  Theorem 2.3 ([3, Theorem 1.8]).

Let $\mathcal{O}$ be a partial ovoid in a Hermitian polar space in a vector
space of dimension $n\geq 4$ over a field with $q=p^{2h}$, where $p$ is prime,
elements.

  1. (a)

We have

$|\mathcal{O}|\leq\left[\binom{n+p-2}{p-1}^{2}-\binom{n+p-3}{p-2}^{2}\right]^{h}+1.$

  2. (b)

If there exists a positive integer $u$ satisfying

$n-2\leq up\leq n+p-4,$

then

$|\mathcal{O}|\leq\left[\binom{n+p-2}{p-1}^{2}-\binom{n+p-3}{p-2}^{2}-\binom{up+1}{n-1}^{2}+\binom{up}{n-1}^{2}\right]^{h}+1.$

##  3\. A Ramsey bound on partial $m$-ovoids

A classic result of Erdős and Szekeres [20] states that
$R(s,t)\leq\binom{s+t-2}{s-1}$, where $R(s,t)$ denotes the smallest number of
vertices $v$ that for any graph $\Gamma$ on $v$ vertices, either $\Gamma$
contains a clique of size $s$ or a coclique of size $t$. For fixed $s$, this
upper bound is roughly $O(t^{s-1})$, which was improved to

$R(s,t)\leq c_{s}\frac{t^{s-1}}{(\log t)^{s-2}}$

by Ajtai, Komlós and Szemerédi [1]. For $s=t$, a recent breakthrough of Campos
et al. [13] showed that

$R(t,t)\leq(4-\epsilon)^{t},$

thus making an exponential improvement over the classic upper bound of
$\binom{2t-2}{t-1}\approx(1+o(1))\frac{4^{t}}{\sqrt{\pi t}}$.

We show that partial $m$-ovoids can give lower bounds for Ramsey numbers, or
equivalently, upper bounds on Ramsey numbers imply upper bounds on the size of
a partial $m$-ovoid.

######  Theorem 3.1.

Let $k$ be the maximum size of a partial $1$-ovoid in a finite polar space
$\mathcal{P}$. Then for any partial $m$-ovoid $\mathcal{O}_{m}$ of
$\mathcal{P}$, we have

$|\mathcal{O}_{m}|<R(m+1,k+1).$

###### Proof.

Let $\Gamma$ be the collinearity graph of $\mathcal{P}$. Then $\Gamma$ does
not contain a coclique of size $k+1$, since $k$ is the maximum size of a
partial ovoid. The induced subgraph $H$ on $\mathcal{O}_{m}$ does not contain
a clique of size $m+1$, as any maximal clique of $\Gamma$ corresponds to a
generator of the polar space and hence meets $\mathcal{O}_{m}$ in at most $m$
points. Therefore $H$ is a graph on $\mathcal{O}_{m}$ vertices with no clique
of size $k+1$ and no coclique of size $m+1$, thus proving the bound. ∎

######  Corollary 3.2.

Let $m\geq 2$ and let $r\geq(m+\log_{p}(2m)^{2})(p-1)+1$. Then a polar space
of rank $r$ over a field of characteristic $p$ does not posses an $m$-ovoid.

###### Proof.

The size of an $m$-ovoid in a polar space of rank $r$ and type $e$ over
$\mathbb{F}_{q}$, with $q=p^{h}$, is $m(q^{r+e-1}+1)=m(p^{hr+he-1}+1)$. As
$p\geq 2$ and $m\geq 2$, we can assume that $r\geq 7$. We want to apply
Theorem 3.1 with $R(m+1,k+1)\leq\binom{m+k}{m}\leq k^{m}$, where
$k=\binom{p+n-1}{p-1}^{h}$. Here we used that $n\geq 2r\geq 14$ and $m\geq 2$.
Hence, it suffices to show that if $r\geq(m+\log_{p}(2m)^{2})(p-1)+1$, then

$\displaystyle m\cdot p^{hr+he-1}\geq\binom{p+n-1}{p-1}^{mh}.$

Taking logarithms and using $n\leq 2r+2$ shows that this is implied by

(1) $\displaystyle r-1\geq m(p-1)\log_{p}(2r+2+p).$

As $r/\log(2r+C)$ grows monotone in $r$, it suffices to show Inequality (1)
for $r=(m+\log_{p}(2m)^{2})(p-1)+1$. We find

$\displaystyle\log_{p}(2r+2+p)$
$\displaystyle\leq\log_{p}((2m+2+\log_{p}(2m)^{2})(p-1))$
$\displaystyle\leq\log_{p}(p-1)+\log_{p}(2m+2+2\log_{p}2m)$ $\displaystyle\leq
1+\log_{p}(2m+2+2\log_{p}2m).$

Hence, to show Inequality 1, it suffices to show that

$\displaystyle(2m)^{2}\geq 2m+2+2\log_{p}2m.$

This is easily verified for $p\geq 2$ and $m\geq 2$. ∎

We note that while the results of [6], [16], and [22] on the non-existence of
$m$-ovoids only apply to the polar spaces
$\operatorname{\mathsf{Q}}^{-}(2r+1,q)$, $\operatorname{\mathsf{W}}(2r-1,q)$,
and $\operatorname{\mathsf{H}}(2r,q^{2})$. The bound in Corollary 3.2 applies
to all finite classical polar spaces.

##  4\. Spectral bounds on partial $2$-ovoids and non-existence of $2$-ovoids

In this section we show the bounds on partial $2$-ovoids that one can obtain
by using spectral arguments and compare them to results based on Ramsey
numbers. We show that while the bound on partial $2$-ovoids are worse, they
allow us to show the non-existence of $2$-ovoids in a wider class of polar
spaces.

Let $P$ be a point of a finite polar space $\mathcal{P}$. We can project the
elements of $\mathcal{P}$ incident with $P$ to (abstractly) obtain a finite
polar space with the same type as $\mathcal{P}$, but with rank one less than
that of $\mathcal{P}$. For instance, each line of $\mathcal{P}$ incident with
$P$ maps to a point of the _quotient_ polar space. By using the non-existence
of ovoids in the quotient polar space, we can rule out the existence of
2-ovoids in the original polar space.

######  Theorem 4.1 ([6, Theorem 14]).

The following polar spaces do not admit $2$-ovoids:

  1. (i)

$\operatorname{\mathsf{W}}(2r-1,q)$ for $r>2$ and $q$ odd;

  2. (ii)

$\operatorname{\mathsf{Q}}^{-}(2r+1,q)$ for $r>2$;

  3. (iii)

$\operatorname{\mathsf{H}}(2r,q^{2})$ for $r>2$;

  4. (iv)

$\operatorname{\mathsf{Q}}(2r,q)$ for $r>4$.

We say that a graph, neither complete nor edge-less, is _strongly regular_
with parameters $(v,k,\lambda,\mu)$ if it has $v$ vertices, degree $k$, each
pair of adjacent vertices $x,y$ have precisely $\lambda$ common neighbours,
and each pair of distinct nonadjacent vertices $x,y$ have precisely $\mu$
common neighbours. The eigenvalues of a strongly regular graph are $k\geq
e^{+}\geq 0>e^{-}$ (cf. [10]). The collinearity graphs on the points of a
polar space are strongly regular graphs. We will use the following well-known
result (see [10, Prop. 1.1.6]).

######  Lemma 4.2.

The induced subgraph on a vertex set $Y$ of a $k$-regular graph $\Gamma$ of
order $v$ with smallest eigenvalue $s$ has average degree at least
$|Y|\frac{k-s}{v}+s$.

Now we can use both the information of the strongly regular graph, and the
size of partial ovoids of a quotient polar space, to bound the size of a
partial 2-ovoid.

######  Theorem 4.3.

Let $r\geq 3$ and let $\mathcal{O}_{2}$ be a partial $2$-ovoid of a finite
polar space $\mathcal{P}$ of rank $r$ and with defining field of order $q$.
Then $|\mathcal{O}_{2}|\leq q\cdot b(n-2,q)+\sigma_{r}$ where $\sigma_{r}$ is
the ovoid number of $\mathcal{P}$, and $b(n^{\prime},q)$ is the maximal size
of a partial ovoid in a polar space of the same type as ${\mathcal{P}}$ in an
$n^{\prime}$-dimensional vector space over the field with $q$ elements.

###### Proof.

First, the collinearity graph $\Gamma$ on the points of $\mathcal{P}$ is a
strongly regular graph. Let $\sigma_{r}$ be the ovoid number of $\mathcal{P}$.
Then the parameters of $\Gamma$ (see [10, §2.5.2]) are

$\displaystyle v=\sigma_{r}\frac{q^{r}-1}{q-1},$ $\displaystyle
e^{+}=q^{r-1}-1,$ $\displaystyle k=q\frac{q^{r-1}-1}{q-1}\sigma_{r-1},$
$\displaystyle e^{-}=-\sigma_{r-1}.$

By Lemma 4.2, we find an element $P\in\mathcal{O}_{2}$ adjacent to at least
$M:=|\mathcal{O}_{2}|\frac{k-s}{v}+s$ vertices. Hence, as no line contains
more than $2$ points, $\mathcal{O}:=\\{\langle P,Q\rangle/P:Q\in Y,Q\in
P^{\perp}\\}$ has size at least $1$. Since $\mathcal{O}$ is a partial
$1$-ovoid of $P^{\perp}/P$, we have

$|\mathcal{O}_{2}|\frac{k-s}{v}\leq b(n-2,q)-s.$

So, by Theorem 2.1, we have

$\displaystyle|\mathcal{O}_{2}|$
$\displaystyle\leq\frac{v}{k-e^{-}}\left(b(n-2,q)-e^{-}\right)$
$\displaystyle=\frac{\sigma_{r}\frac{q^{r}-1}{q-1}}{(q\frac{q^{r-1}-1}{q-1}+1)\sigma_{r-1}}\left(b(n-2,q)+\sigma_{r-1}\right)$
$\displaystyle=\frac{\sigma_{r}}{\sigma_{r-1}}b(n-2,q)+\sigma_{r}.$

Now $\sigma_{r}=q\sigma_{r-1}-q+1$ and so $|\mathcal{O}_{2}|\leq q\cdot
b(n-2,q)+\sigma_{r}$. ∎

Together with Theorem 2.1, Theorem 2.2, and Theorem 2.3, we have the
following:

######  Corollary 4.4.

Let $r\geq 3$ and let $\mathcal{O}_{2}$ be a partial $2$-ovoid of a finite
polar space $\mathcal{P}$ of rank $r$ and with defining field of order $q$.
Then

$|\mathcal{O}_{2}|\leq\begin{cases}q^{r}+q+1+q\left[\binom{p+2r-3}{p-1}\right]^{h}&\text{if
}\mathcal{P}\cong\operatorname{\mathsf{W}}(2r-1,q),\\\
q^{r-\varepsilon}+q+1+q\left[\binom{n+p-4}{p-1}-\binom{n+p-6}{p-3}\right]^{h}&\text{if
}\mathcal{P}\cong\operatorname{\mathsf{Q}}^{\varepsilon}(2r-1,q),\\\
q^{2r+1}+q^{2}+1+q^{2}\left[\binom{n+p-4}{p-1}^{2}-\binom{n+p-5}{p-2}^{2}\right]^{h}&\text{if
}\mathcal{P}\cong\operatorname{\mathsf{H}}(n-1,q^{2}),n\text{ odd},\\\
q^{2r-1}+q^{2}+1+q^{2}\left[\binom{n+p-4}{p-1}^{2}-\binom{n+p-5}{p-2}^{2}\right]^{h}&\text{if
}\mathcal{P}\cong\operatorname{\mathsf{H}}(n-1,q^{2}),n\text{
even}.\end{cases}$

Note that Theorem 4.1 is also implied by Corollary 4.4.

######  Corollary 4.5.

Let $r\geq(1+\log_{p}(7))p+1$ for $p\geq 5$, or $r\geq 6$ for $p=2,3$. Then a
polar space of rank $r$ over a field of characteristic $p$ does not posses a
$2$-ovoid.

###### Proof.

The size of a $2$-ovoid in a polar space of rank $r$ and type $e$ over
${\mathbb{F}}_{q}$, with $q=p^{h}$, is
$2(q^{r+e-1}+1)=(q^{r+e-1}+1)+(p^{hr+he-1}+1)$. Note that the first summand
also occurs in the bounds in Corollary 4.4, so we only need to show that
$p^{hr+he-1}+1$ is larger than $q\binom{p+2r-3}{p-1}^{h}+q-1$.

First we show the claim for $p\geq 3$. Using $p-1\geq 2$, we have

$\displaystyle q\binom{p+2r-3}{p-1}^{h}+q-1\leq
p^{h}\cdot(2r+e+p)^{h(p-1)}\leq(2r+e+p)^{h(p-1)+1}.$

We want to show that

$\displaystyle p^{hr+he-1}>(2r+e+p)^{h(p-1)+1}$

as then $2$-ovoids cannot exist. This is equivalent to

$\displaystyle(h(r-p+1)-2)+(h(p-1)+1)=hr+he-1>(h(p-1)+1)\log_{p}(2r+e+p).$

We can bound the left-hand side by $hr+he-1\geq h(r-1)$ and the ride-hand side
by $(h(p-1)+1)\log_{p}(2r+e+p)\leq hp\log_{p}(2r+p+2)$. Hence, it suffices to
show that

$r-1>p\log_{p}(2r+p+2).$

For $r=(1+\log_{p}(7))p+1$, we find

$\displaystyle\log_{p}(2r+p+2)=\log_{p}((3+2\log_{p}(7))p+4)\leq\log_{p}((4+2\log_{p}(7))p)=1+\log_{p}(4+2\log_{p}(7)).$

As $2\log_{p}(7)<3$ for $p\geq 5$, this shows $r-1>p\log_{p}(2r+p+2)$.

For $p=2,3$, it can be verified that $r\geq 6$ suffices. ∎

Corollary 3.2 gives the condition of the form
$r\geq(2+\log_{p}(4)^{2})(p-1)+1$ which is worse than Corollary 4.5 for $p\geq
3$.

##  5\. Constructions

In this section we give a general probabilistic construction of partial
$m$-ovoids, when $m$ grows at least linearly with respect to the rank $r$ of
the polar space, and an explicit construction for the special case of $m=2$ in
the binary symplectic polar space, via an equivalence with _nearly orthogonal
sets_. We also give constructions of partial $m$-ovoids for $m>2$ that have a
probabilistic component in them.

###  5.1. Probabilistic construction

In this section we give a general probabilistic construction of partial
$m$-ovoids, when $m$ grows at least linearly with respect to the rank $r$ of
the polar space.

######  Proposition 5.1.

In a finite polar space of rank $r$, type $e$, over a finite field of order
$q$, there exists a partial $m$-ovoid of size

$q^{\Omega(r)},$

for every $m\geq 0.51r$.

###### Proof.

Let $\mathcal{P}$ be a finite polar space of rank $r$ and type $e$ over the
finite field $\mathbb{F}_{q}$. Let $N=\theta_{r}(q^{r+e-1}+1)$ be the total
number of points in $\mathcal{P}$, where $\theta_{r}=(q^{r}-1)/(q-1)$ and let
$M=\prod_{i=1}^{r}(q^{i+e-1}+1)$ be the total number of generators. Pick a
random subset $S$ of $\mathcal{P}$ by picking each point independently with
probability $\rho$, where the exact value of $\rho$ will be specified later.
For a fixed generator, the probability that $S$ contains at least $m+1$ points
from it is

$\binom{\theta_{r}}{m+1}\rho^{m+1}.$

Therefore, by the union bound (for probabilities) over all generators, if we
have $M\binom{\theta_{r}}{m+1}\rho^{m+1}<1/2$ then with probability $>1/2$,
$S$ does not contain $m+1$ points from any generator, that is, it is a partial
$m$-ovoid. Since $M=(1+o(1))q^{r(r+1)/2+r(e-1)}$ and
$\binom{\theta_{r}}{m+1}<q^{(r-1)(m+1)}/2$, taking

$\rho=q^{-r(r+1)/(2(m+1))-r(e-1)/(m+1)-r+1}$

suffices to get the inequality. The expected size of $S$ is $\rho
N=q^{r-r(r+1)/(2(m+1))-r(e-1)/(m+1)}$, and with probability $\geq 1/2$ there
exists an $S$ with size at least $\lfloor\rho N\rfloor$, since we have a
binomial distribution. Therefore, with a positive probability there exists a
set of size at least $\lfloor q^{r-r(r+1)/(2(m+1))-r(e-1)/(m+1)}\rfloor$ that
meets every generator in at most $m$ points. In particular, for $m=0.51r$, we
have the existence of a partial $m$-ovoid of size $q^{\Omega(r)}$. ∎

###  5.2. Oddtown, nearly orthogonal sets, and triangle-free graphs

A family $\mathcal{F}$ of subsets of a $t$-element set is an _Oddtown_ if
$|S|$ is odd for every $S\in\mathcal{F}$, and $|S\cap T|$ is even for every
pair of distinct elements $S,T\in\mathcal{F}$. It was shown by Berlekamp [8],
that the size of an Oddtown is at most $t$ using a simple linear algebraic
argument. There is a direct connection between partial $m$-ovoids of binary
symplectic spaces and similar families of subsets. Let $V$ be the
$2r$-dimensional vector subspace of $\mathbb{F}_{2}^{2r+1}$ consisting of all
even weight vectors, i.e.,
$V=\\{(x_{1},\dots,x_{2r+1}:\sum_{i=1}^{2r+1}x_{i}=0\\}$. Then $V$ along with
the “standard” dot product,
$(x_{1},\dots,x_{2r+1})\cdot(y_{1},\dots,y_{2r+1})=\sum_{i=1}^{2r+1}x_{i}y_{i}$
gives rise to a symplectic polar space isomorphic to
$\operatorname{\mathsf{W}}(2r-1,2)$. Let $\mathcal{O}$ be a partial $m$-ovoid
in this model of $\operatorname{\mathsf{W}}(2r-1,2)$. Then the vectors
representing elements of $\mathcal{O}$ have the following properties:

  1. (i)

For all $u\in\mathcal{O}$, $x\cdot x=0$.

  2. (ii)

For any $(m+1)$ distinct vectors $x_{1},\dots,x_{m+1}\in\mathcal{O}$ there
exists $i\neq j$ such that $x_{i}\cdot x_{j}=1$.

If we think of vectors here as characteristic vectors of subsets of
$\\{1,\ldots,2r+1\\}$, then in terms of set families, we have a family
$\mathcal{F}\subseteq 2^{[2r+1]}$ such that for all $S\in\mathcal{F}$, $|S|$
is even, and for all distinct $S_{1},\dots,S_{m+1}\in\mathcal{F}$, there
exists an $i\neq j$ such that $|S_{i}\cap S_{j}|$ is odd. By taking
complements of these sets, we obtain the following generalisation of an
Oddtown:

######  Definition 5.2 (Generalised Oddtown).

A family $\mathcal{G}$ of subsets of a $t$-element set is a _generalised
Oddtown_ , with parameter $m$, if:

  1. (i)

for all $S\in\mathcal{G}$, $|S|$ is odd;

  2. (ii)

for all distinct $S_{1},\dots,S_{m+1}\in\mathcal{G}$, there exists an $i\neq
j$ such that $|S_{i}\cap S_{j}|$ is even.

When $m=1$, we have an Oddtown.

Let $\Gamma_{t}$ be the graph whose vertices are the proper subsets of
$\\{1,\ldots,t\\}$ of odd cardinality, and two such subsets are adjacent if
they are not equal and have intersection of odd cardinality. For $t$ odd,
$\Gamma_{t}$ is isomorphic to the collinearity graph of the symplectic space
$\operatorname{\mathsf{W}}(t,2)$, and for $t$ even, $\Gamma_{t}$ is isomorphic
to the induced subgraph of the collinearity graph, on the points of
$\operatorname{\mathsf{W}}(t-1,2)$ not lying in a fixed hyperplane. Therefore,
the clique number of $\Gamma_{t}$ is $2^{\lfloor(t-1)/2\rfloor}$. A
generalised Oddtown with parameter $m$ translates to a subset of the vertices
of $\Gamma_{t}$ not containing an $(m+1)$-clique. So if $m>2^{(t-1)/2}$, then
the set of all subsets of $\\{1,\ldots,t\\}$ of odd cardinality is a
generalised Oddtown.

To avoid unnecessary case distinctions depending on the parity of $t$, we will
assume that $t$ is odd and write $\operatorname{\mathsf{W}}(t,2)$. If $t$ is
even, then everything we write holds for the graph obtained above from
$\operatorname{\mathsf{W}}(t-1,2)$. Recall that a subset ${\mathcal{F}}$ of
vectors of ${\mathbb{F}}^{t}$ is called $m$-nearly orthogonal if its elements
are not self-orthogonal and every subset of $m+1$ members includes an
orthogonal pair. For ${\mathbb{F}}={\mathbb{F}}_{2}$, it is clear from the
discussion above that this is equivalent to our generalised Oddtown problem as
well as to the maximal size of a partial $m$-ovoid in
$\operatorname{\mathsf{W}}(t,2)$. Thus, the results of [21] on nearly
orthogonal sets imply that that there exists a partial $2$-ovoid of
$\operatorname{\mathsf{W}}(t,2)$ of size $t^{1+\delta}$ for some constant
$\delta>0$. In particular, the generalised Oddtown problem with parameter $2$
does not have a linear bound. More recently, it has been shown in Theorem 1.1
of [14] that there exists a constant $\delta>0$ such that for all integers
$t\geq m\geq 2$, there exists a partial $m$-ovoid of
$\operatorname{\mathsf{W}}(t,2)$ of size at least

$t^{\delta m/\log m}.$

Note that this does not imply anything for small $m$. In all of the
aforementioned results, $\delta$ can be calculated without too much effort
from the respective proofs. For instance, the proof of [14, Theorem 1.1] shows
that we find a partial $800$-ovoid of size $100^{1.25\ell-C}$ in
$\operatorname{\mathsf{W}}(100^{\ell},2)$.

Here we give an explicit construction for partial $2$-ovoids that improves
these previous results:

######  Proposition 5.3.

There exists a $2$-nearly orthogonal set of size at least $t^{1.12895}$ in
${\mathbb{F}}_{2}^{t}$ that can be constructed explicitly.

###### Proof.

Consider the Cayley graph $\Gamma=\mathrm{Cay}(G,S)$ with
$G=\mathbb{F}_{2^{h}}^{2}$ and
$S=\\{(a,a^{3}):a\in{\mathbb{F}}_{2^{h}}^{2}\setminus\\{0\\}\\}$. In Theorem 2
of [17] it is shown that the adjacency matrix of the complement
$\overline{\Gamma}$ of $\Gamma$ has $2$-rank at most

$\tau:=\frac{1+\sqrt{2}}{2}(2+\sqrt{2})^{h}.$

Following the proof of Theorem 1.11 in [21], this implies that we can embed
$\overline{\Gamma}$ in ${\mathbb{F}}_{2}^{\tau}$. As $\Gamma$ is triangle-
free, see [17, §2.3], this embedding corresponds to a $2$-nearly orthogonal
set. Hence, we obtain the asserted bound (note:
$\log(4)/\log(2+\sqrt{2})\approx 1.12895$). ∎

######  Corollary 5.4.

There exists a partial $2$-ovoid of $\operatorname{\mathsf{W}}(t,2)$ of size
at least $t^{1.12895}$ that can be constructed explicitly.

In [7, Theorem 4.6] we find the following result, where we note that their
proof works for any field, not only the reals:

######  Theorem 5.5.

Let $m>1$ be a positive integer. Let $\Gamma$ be a graph on $n$ vertices and
$r:=\mathrm{rank}_{\mathbb{F}}(A_{\Gamma}+I)$ for some field ${\mathbb{F}}$.
Then, for every large enough $h$, there exists a graph $\Delta$ such that:

  1. (1)

the clique number of $\Delta$ is at most $m$,

  2. (2)

$\mathrm{rank}_{\mathbb{F}}(A_{\Delta}+I)\leq r^{h}$,

  3. (3)

the graph $\Delta$ has

$\frac{1}{m}\left(\frac{n}{\left(\sum_{t=1}^{m}t^{m+1}a_{i}\right)^{\frac{1}{m+1}}}\right)^{h}-1$

vertices. Here $a_{i}$ denotes the number of cliques of size $i$ in $\Gamma$.

Using this result and a $32$-vertex triangle-free graph with low complementary
rank, we prove the following.

######  Proposition 5.6.

For sufficiently large $t$, the following is true:

  1. (1)

There exists a $28$-clique-free graph with complementary $2$-rank $t$ and
$t^{1.12936}$ vertices.

  2. (2)

There exists a $42$-clique-free graph with complementary $2$-rank $t$ and
$t^{1.15392}$ vertices.

  3. (3)

For $\varepsilon>0$, there exists an $m$ such that there is an infinite family
of $m$-clique-free graphs with complementary $2$-rank $t$ and
$t^{\frac{\log(16)}{\log(10)}-\varepsilon}$ vertices.

###### Proof.

Consider the following vectors $S$ of ${\mathbb{F}}_{2}^{5}$:

$\displaystyle(1,0,0,0,0),(0,1,0,0,0),(0,0,1,0,0),(0,0,0,1,0),(0,0,0,0,1),$
$\displaystyle(1,1,1,0,0),(1,1,0,1,0),(1,0,1,1,0),(0,1,1,1,1).$

As homogeneous coordinates of projective points, they form a _cap_ $\Sigma$ of
$\mathrm{PG}(4,2)$: a set of points with no 3 collinear. Now we construct the
_linear representation_ $L(\Sigma)$ of $\Sigma$, a vertex-transitive graph, as
follows. The vertices are the vectors of ${\mathbb{F}}_{2}^{5}$, and we join
$u$ and $v$ by an edge if their difference $u-v$ lies in $S$. Since $S$ is
inverse-closed (because our vectors are over ${\mathbb{F}}_{2}$), it follows
that $L(\Sigma)$ is undirected. Moreover, since $\Sigma$ is a cap, it follows
that $L(\Sigma)$ is regular of degree $|S|$ and triangle-free. It has 32
vertices and 144 edges. Furthermore, if $A$ is the adjacency matrix of
$L(\Sigma)$, then $\mathrm{rank}_{2}(A+I)=10$. Hence, we can apply Theorem 5.5
with $a_{1}=32$ and $a_{2}=144$. For $m=28$, Theorem 5.5 gives a graph
$\Delta$ with

$\frac{1}{28}\cdot 13.48011787424788^{h}$

vertices and $\mathrm{rank}_{2}(A_{\Delta}+I)=10^{h}$. The calculation for
$m=42$ is similar. The limit of

$\frac{32}{(32+144\cdot 2^{m+1})^{\frac{1}{m+1}}}$

for $m\rightarrow\infty$ is $16$ which shows the last claim. ∎

We note that $\log(16)/\log(10)>1.20412$.

######  Corollary 5.7.

For $t$ sufficiently large, $\operatorname{\mathsf{W}}(t,2)$ possesses a
partial $28$-ovoid of size $t^{1.2936}$ and a partial $42$-ovoid of size
$t^{1.15392}$.

######  Remark 5.8.

  1. (i)

The 9-regular graph $\mathcal{D}$ on 32 vertices in the proof of Proposition
5.6 was the best found via a large search for triangle-free graphs with low
complementary 2-rank. It was found by taking graphs of homogeneous coherent
configurations of order at most 33 in the database provided by [5]. We also
considered graphs of Schurian association schemes of primitive groups of
degree at most 80, with permutation rank at most 6. No ‘better’ graph has been
found than $\mathcal{D}$, so far, in terms of $\log_{r}(n)$, where $n$ is the
order and $r$ is the complementary 2-rank.

  2. (ii)

The spectrum of $\mathcal{D}$ is

$9^{(1)},5^{(1)},1^{(21)},(-3)^{(7)},(-7)^{(2)}.$

Therefore, using the graph $\mathcal{D}$ in the place of $\mathcal{C}$ in [7,
Corollary 4.5(1)], we find that $\nu_{2}(32\ell)\leq 11\ell+21$, where
$\nu_{d}(n)$ is the smallest rank of $A_{G}+I$ over $d$-clique-free graphs $G$
of order $n$. Therefore, $\nu_{2}(n)\leq(11/32+o(1))n$. This is an improvement
on the bound in [7, §4] where it was shown that $\nu_{2}(n)\leq(3/8+o(1))n$.
In particular, the Rank-Ramsey number $R^{k}(3,n)$ satisfies
$R^{k}(3,11\ell+22)>32\ell$, for every $\ell>2$. We minimize
$\frac{11\ell+22}{32\ell-\log 2}$ at $\ell=38$. Hence, this improves the
exponent in [7, Theorem 4.1(2)] from $\log_{296}(232)\approx 0.957$ to
$\log_{608}(440)\approx 0.950$. That is, we obtain that
$\nu_{d}(n)=O\left(n^{\log_{608}(440)}\right)$ for sufficiently large $d$.

##  6\. Conclusion

We have established a new connection between partial $m$-ovoids in finite
classical polar spaces and Ramsey numbers (Theorem 3.1). It is conceivable
that this connection can lead to new explicit constructions for various Ramsey
numbers. Of particular interest is the case when $m$ is linear in the rank of
a polar space, as it can be used to show that $R(t,t)>\mathrm{exp}(t)$ via an
explicit construction (see Proposition 5.1 which shows the existence of these
objects), which is one of the main open problems in Ramsey theory. Our
connection to partial $m$-ovoids can also lead to improved lower bounds on
various small Ramsey numbers.

Our constructions in the binary symplectic space use graphs with certain rank
properties to construct large partial $m$-ovoids. It will be of interest to
look for similar constructions in other finite classical polar spaces. For
others fields we lose the direct connection to nearly orthogonal sets. Perhaps
it is still possible to use the construction in [23] on nearly orthogonal sets
over arbitrary fields. We also believe that the probabilistic construction can
be improved further.

Finally, it would be very interesting to improve the upper bounds on partial
$m$-ovoids or prove their asymptotic tightness for some values of $m$ by
giving better constructions.

### Acknowledgements

The third author was supported by a postdoctoral fellowship of the Research
Foundation – Flanders (FWO). The first author thanks Delft University of
Technology for their hospitality while he was on sabbatical there, where most
of this work was done. We thank Peter Frankl for discussing the Oddtown
Theorem and its possible generalisations with us. We also thank Sam Mattheus
for various fruitful discussions.

## References

  * [1] M. Ajtai, J. Komlós, and E. Szemerédi, A note on Ramesy numbers, J. Comb. Theory Ser. A, 29 (1980) 354–360.
  * [2] N. Alon, A. Shpilka, and C. Umans, On sunflowers and matrix multiplication, Comput. Complexity 22 (2013), 219–243.
  * [3] O. Arslan and P. Sin, Some simple modules for classical groups and p-ranks of orthogonal and Hermitian geometries, J. Algebra 327 (2011) 141–169.
  * [4] S. Ball and V. Pepe, Asymptotic improvements to the lower bound of certain bipartite Turán numbers, Combin. Probab. Comput., 21 (3), (2012), 323–329.
  * [5] J. Bamberg, A. Hanaki, J. Lansdown, AssociationSchemes, a GAP package for working with association schemes and homogeneous coherent configurations, Version 3.0.0. http://www.jesselansdown.com/AssociationSchemes, GAP package, 2023.
  * [6] J. Bamberg, S. Kelly, M. Law, and T. Penttila, Tight sets and $m$-ovoids of finite polar spaces, J. Combin. Theory Ser. A 114 (2007) 1293–1314.
  * [7] G. Beniamini, N. Linial, and A. Schraibman, The Rank-Ramsey Problem and the Log-Rank Conjecture, https://arxiv.org/abs/2405.07337
  * [8] E. R. Berlekamp, On subsets with intersections of even cardinality, Canad. Math. Bull. 12 (1969) 471–474.
  * [9] A. Blokhuis and G. E. Moorhouse, Some $p$-ranks related to orthogonal spaces, J. Alg. Combin. 4 (1995) 295–316.
  * [10] A. E. Brouwer and H. Van Maldeghem, Strongly Regular Graphs, Cambridge University Press, Encyclopedia of Mathematics and Its Applications 182, 2022.
  * [11] F. Buekenhout and E. Shult, On the foundations of polar geometry, Geometriae Dedicata 3 (1974), 155–170.
  * [12] R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc. 18 (2) (1986), 97–122.
  * [13] M. Campos, S. Griffiths, R. Morris, and J. Sahasrabudhe, An exponential improvement for diagonal Ramsey, arXiv preprint (2023) https://arxiv.org/abs/2303.09521.
  * [14] D. Chawin and I. Haviv, Nearly orthogonal sets over finite fields, arXiv preprint (2024) https://arxiv.org/abs/2402.08274v1.
  * [15] D. Conlon and A. Ferber, Lower bounds for multicolor Ramsey numbers, Adv. Math. 378 (2021), 107528.
  * [16] J. De Beule, J. Mannaert, and V. Smaldore, Some non-existence results on $m$-ovoids in classical polar spaces, European J. Combin., 118 (2024), 103943.
  * [17] H. Deng, H. Huang, G. Weng, and Q. Xiang, The BCH family of storage codes on triangle-free graphs is of unit rate, 2023. arxiv preprint https://arxiv.org/abs/2310.04820.
  * [18] J. Eisfeld, On the common nature of spreads and pencils in ${PG}(d,q)$, Discrete Math., 189(1-3):95–104, 1998.
  * [19] J. S. Ellenberg and D. Gijswijt, On large subsets of $\mathbb{F}_{q}^{n}$ with no three-term arithmetic progression, Ann. of Math. 185 (2) (2017), 339–343.
  * [20] P. Erdős, G. Szekeres. A combinatorial problem in geometry, Compos. Math. 2 (1935) 463–470.
  * [21] A. Golovnev and I. Haviv, The (generalized) orthogonality dimension of (generalized) Kneser graphs: Bounds and applications, Theory of Computing, 18 (2022), 1–22.
  * [22] N. Hamilton and R. Mathon, Existence and non-existence of $m$-systems of polar spaces, European J. Combin., 22(1):51–61, 2001.
  * [23] I. Haviv, S. Mattheus, A. Miloević, and Y. Wigderson. Larger nearly orthogonal sets over finite fields, arxiv preprint (2024) https://arxiv.org/abs/2404.01057.
  * [24] R. Hill, Caps and codes, Discrete Math. 22 (2) (1978), 111–137.
  * [25] J. W. P. Hirschfeld and J. A. Thas.  General Galois geometries.  Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1991.  Oxford Science Publications.
  * [26] S. Mattheus and J. Verstraete. The asymptotics of $r(4,t)$, Ann. Math., 199 (4) (2024).
  * [27] B. Segre, Forme e geometrie hermitiane, con particolare riguardo al caso finito, Ann. Mat. Pura Appl. 70 (4) (1965), 1–201.
  * [28] E. E. Shult and J. A. Thas, $m$-Systems and partial $m$-systems of polar spaces, Des. Codes Cryptogr. 8 (1996) 229–238.
  * [29] J. A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1981), 135–143.
  * [30] J. Thas, Interesting pointsets in generalised quadrangles and partial geometries. Linear Algebra Appl., 114-115 (1989), 103 – 131.
  * [31] J. Thas and H. van Maldeghem, Suzuki-Tits ovoids through the years. Arch. Math. 121 (2023), 511 – 521.
  * [32] J. Tits, Sur la trialité et certains groupes qui s’en déduisent, Inst. Hautes Études Sci. Publ. Math., (2):13–60, 1959.
  * [33] J. Tits, Ovoïdes à translations, Rend. Mat. e Appl. 21 (5) (1962), 37–59.
  * [34] J. Tits, Ovoïdes et groupes de Suzuki, Arch. Math. 13 (1962), 187 – 198.