Ryser’s conjecture

I am on a research visit in Rome, working with Valentina Pepe, and our joint paper on Ryser’s conjecture is on arXiv now. So this seems like the right time to talk about the conjecture and the problems related to it that I have been obsessing over for past few months.

In graph theory, the classical result of Kőnig says that in any (finite) bipartite graph the minimum number of vertices that cover all edges (the so-called vertex cover number) is equal to the maximum number of pairwise disjoint edges in the graph (matching number). This result is clearly not-true for non-bipartite graphs (think of some obvious counter examples), where the best one can do is have the vertex cover number equal to twice the matching number since the vertices contained in any maximum matching must cover all the edges. Ryser’s conjecture is a proposed generalisation of this result to $r$-partite hypergraphs, that is, for hypergraphs whose vertex set can be partitioned into $r$ parts (let’s call them sides from now on), such that each edge of the hypergraph contains a unique vertex from each of the sides. In particular, every $r$-partite hypergraph is $r$-uniform, that is, each edge has exactly $r$ vertices in it. If we have any $r$-uniform hypergraph $\mathcal{H}$ (not necessarily $r$-partite), then the following follows as before, $\tau(\mathcal{H}) \leq r \nu(\mathcal{H})$, where $\tau$ denotes the vertex cover number and $\nu$ the matching number. Ryser’s conjectured that if $\mathcal{H}$ is $r$-partite, then we must have $\tau(\mathcal{H}) \leq (r - 1) \nu(\mathcal{H})$. This conjecture first appeared in the Ph.D. thesis of Ryser’s student, J.R. Henderson, and it has often been misattributed to a 1967 paper of Ryser (see this).

Despite the time span of about 50 years, our current knowledge about this conjecture is abysmal. The only other case of this conjecture which is known to be true in general is $r = 3$, which was proved by Aharoni using topological methods! If we impose some further restrictions on the hypergraph, then we know a bit more, but not much. The conjecture is true for intersecting hypergraphs (matching number equal to $1$) if $r \leq 5$, as proved by Tuza, and for $r \leq 9$ if the hypergraph is also linear, as proved in the recent paper by Francetić, Herke, McKay and Wanless.

In view of our inability to prove this conjecture, some natural questions to ask are, (1) is it even true?, and (2) why is it so hard to prove?. While it’s quite possible that the conjecture is false, let’s focus on (2) for now. Sometimes what makes an extremal problem in combinatorics hard to prove is that there are many different kinds of extremal examples, and a “combinatorial” proof must somehow consider all of these examples (I should thank Tibor Szabó for this intuition). So let’s see what we know about hypergraphs meeting the bound in Ryser’s conjecture.

Until recently, the only known family of $r$-partite hypergraphs with vertex cover number equal to $r - 1$ times the matching number, called $r$-Ryser hypergraphs, came from finite projective planes of order $r - 1$ (which is what sparked my interest in this problem), and hence they were known to exist whenever $r - 1$ is equal to a prime power ($2, 3, 4, 5, 6, 8, 9, 10, \dots$). Once you know the definition of projective planes, the construction is easy: remove a point and all lines through it. These hypergraphs are hence known as truncated projective planes. Note that this gives us intersecting $r$-Ryser hypergraphs, and to get such hypergraphs with matching number $\nu$ one can simply take $\nu$ disjoint copies of the intersecting hypergraphs (more on this later!). Inspired by the lack of examples, several people gave constructions for small values of $r$ where projective planes did not exist (or were not known to exist), and then Abu-Khazneh, Barát, Pokrovskiy and Szabó, came up with a clever construction which gave a new infinite family of intersecting $r$-Ryser hypergraphs whenever $r - 2$ is a prime power. In fact, they were able to construct many non-isomorphic examples of such hypergraphs! Note that while it is known that there are plenty of non-isomorphic projective planes of a given order, it is not clear what the rate of growth of this function is, and that’s a fascinating problem on its own. Another interesting family of $r$-Ryser hypergraphs was obtained by Haxell and Scott, whenever both $(r - 1)/2$ and $(r + 1)/2$ are prime powers (whether this gives infinitely many new values of $r$ for which we have a Ryser hypergraph or not is in fact related to a nice open problem in number theory, which a careful reader should be able to deduce :)). Both of these constructions rely on finite projective or affine planes.

Valentina and I have constructed new infinite families of non-intersecting $r$-Ryser hypergraphs, whenever $r - 1$ is a prime power bigger than $3$, which looks fundamentally different from just taking disjoint copies of intersecting Ryser hypergraphs. The condition on $r$ being at least $4$ cannot be relaxed since a result of Haxell, Narins and Szabo, says that every $3$-Ryser hypergraph is essentially obtained by taking disjoint copies of intersecting $3$-Ryser hypergraphs. For $r = 4$ it was already known that such a characterisation cannot be true, because of a computer-generated example of Abu-Khazneh. Our constructions show that for any $r \geq 4$ and $\nu > 1$, such that $r - 1$ is a prime power, there exists an $r$-Ryser hypergraph with matching number equal to $\nu$ which does not contain two vertex disjoint copies of intersecting $r$-Ryser hypergraphs.

The constructions are again finite geometric, and we were quite happy about the fact that many non-trivial results on blocking sites of finite projective planes came into play when proving that these hypergraphs are Ryser extremal. Here is a description of the first family, with $\nu = 2$, along with a picture:

Let $\pi_1$ and $\pi_2$ be two copies of classical projective planes of order $q$ with a common point $P$, which will be truncated at the the points $Q_1$ and $Q_2$. Let $\mathcal{C}$ be a conic in the second plane passing through $q$, and $v$ an extra vertex. The edges of the hypergraph are (1) all lines in the first plane not through $Q_1$ or $P$, and a line $\ell$ through $P$ which does not contain $Q_1$; (2) all the lines of the second projective plane not through $Q_2$ that contain a point of the conic $\mathcal{C}$; (3) two new (weird) edges $e_1$ and $e_2$ with $e_1 = \ell \setminus \{P\} \cup \{R\}$ and $e_2 = C \setminus \{Q_2\} \cup \{v\}$.

The fact that this is a $(q + 1)$-Ryser hypergraph of matching number $2$ with the required properties, whenever $q$ is an odd prime, is proved in the paper. For other values of $q$, there is a second, more involved, construction (which also comes with a picture!).

Our new constructions show that the non-intersecting Ryser hypergraphs can have a richer structure, and perhaps it’ll be useful to construct more such hypergraphs for either disproving the conjecture or understanding the extremal cases better. Some other interesting questions, that have also been mentioned before by others, are as follows:

Open problem 1: Find the minimum number of edges an $r$-Ryser hypergraph can have. It is not known, but conjectured, that linearly many edges should suffice.

Open problem 2: What is the largest vertex cover number of an $r$-partite intersecting hypergraph, if the “trivial” covers containing a side or an edge are not allowed?

This seems to be related to the problem of finding the smallest non-trivial blocking set in finite projective planes.

Open problem 3: Prove, or disprove!, Ryser’s conjecture.

Wenger graphs

A central (and foundational) question in extremal graph theory is the forbidden subgraph problem of Turán, which asks for the largest number of edges in an $n$-vertex graph $G$ that does not contain any copy of a given graph $H$ as its subgraph. This number is denoted by $ex(n, H)$ and it is called the Turán number of the graph $H$. While the Erdős-Stone theorem has solved this problem asymptotically whenever $H$ is non-bipartite, the case of bipartite graphs is still wide open. For example, when $H$ is isomorphic to the cube graph $Q_3$, all we know is that $c_1 n^{3/2} \leq ex(n, Q_3) \leq c_2 n^{8/5}$, for some constants $c_1, c_2$ and large enough $n$. The two most important cases in this problem are when $H$ is a complete bipartite graph or an even cycle. In this post we will focus on the latter (see this for a survey), where graphs derived from finite geometries give us the best known extremal constructions for small cycles.

It was proved by Bondy and Simonovits that $ex(n, C_{2k}) = O(n^{1 + 1/k})$, but this bound is known to be (asymptotically) sharp only for the case of $k = 2, 3$ or $5$; so in particular we do not even know what $ex(n, C_{8})$ is. The sharpness for these cases follows from the existence of finite generalized $n$-gons of order $q$ for every prime power $q$ and $n = 3, 4, 6$. These are point-line geometries introduced by Jacques Tits in $1959$, and an easy graph theoretic definition of these objects is as follows:

A generalized $k$-gon of order $q$, for $k,q \geq 2$, is a point-line geometry whose incidence graph is $(q+1)$-regular, has diameter $k$ and girth $2k$.

One can count the total number of points/lines in a generalized $k$-gon in terms of the parameter $q$, which tells us that the incidence graph has $\Theta(q^{k-1})$ vertices and $\Theta(q^{k})$ edges. Since by definition, such a graph is $C_{2k - 2}$ free, we get examples of $C_{2k - 2}$-free graphs on $n$ vertices with $\Theta(n^{1 + 1/(k-1)})$ edges, whenever such a structure exists. Sadly, it is known that such generalized $k$-gons can only exist for $k \in \{3, 4, 6\}$, and in these cases they are only known to exist when $q$ is a power of a prime. Using the density of prime numbers these objects can be used to prove that $ex(n, 2k) = \Theta(n^{1 + 1/k)})$ for $k = 2, 3, 5$.

The $k = 3$ case is simply a finite projective plane of order $q$, and Tits had already shown the existence (and many examples) for the other cases. Some special cases of generalized $4$-gons and $6$-gons were rediscovered by Benson in 1966 (and in the combinatorics community sometimes he’s the one who is cited for constructing these extremal graphs).

While it’s quite easy to construct generalized $4$-gons of order $q$ using zeros of a non-degenerate quadratic form in the $4$-dimensional projective space over $\mathbb{F}_q$ (these objects are a special case of polar spaces), the construction of a generalized $6$-gon is quite involved. As an attempt to simplify this situation, in 1991 Wenger constructed a family of bipartite graphs $H_k(q)$ for integers $k \geq 2$ and prime power $q$, with $2q^k$ vertices and $q^{k + 1}$ edges, such that the graphs $H_2(q)$, $H_3(q)$ and $H_5(q)$ did not have any $C_4$, $C_6$ and $C_{10}$, respectively. His construction and the proof of the cycle freeness involved some simple algebra over the finite fields. Later on his graphs were studied extensively, from various perspectives, and it was realised (I am not sure exactly when, but it’s at least mentioned here) that $H_3(q)$ is in fact just an induced subgraph of the incidence graph of a well-known generalized quadrangle (the quadric $Q(4,q)$) and $H_5(q)$ is isomorphic to a homomorphic image of the incidence graph of the only known generalized hexagon of order $q$, the split Cayley hexaon. (From the definition of $H_2(q)$ it will be pretty clear that it is a $q$-regular induced subgraph of the incidence graph of the Desarguesian projective plane.)

In this post, we will see how $H_3(q)$ is directly related to a particular family of generalized quadrangles introduced by Tits (which first appeared in Debowski, 1968), known as the $T_2(O)$ generalized quadrangles. These quadrangles are in fact isomorphic to $Q(4,q)$ when $q$ is odd, or in general when $O$ is an irreducible conic (which will be the case corresponding to Wenger graphs). Let’s start with the definition of Wenger graphs.

Construction 1: Let $P$ and $L$ be two copies of $\mathbb{F}_q^k$. Then $H_{k}(q)$ is the bipartite graph defined on $P$ and $L$ by making two vertices $p = (p_1, \dots, p_k)$ and $\ell = (\ell_1, \dots, \ell_k)$ adjacent if the following equations are satisfied:

$p_2 + \ell_2 = p_1 \ell_1$
$p_3 + \ell_3 = p_1 \ell_1^2$
$\dots$
$p_k + \ell_k = p_1\ell_1^{k-1}$

The original equations used by Wenger were a bit different, but it can be (and has been) shown that the graph we get is the same. One of the first thing we notice in these defining relations is that once you have fixed $(p_1, p_2, \dots, p_k)$, every value of $\ell_1$ uniquely determine the vector $(\ell_2, \dots, \ell_k)$, and vice versa. We can thus conclude that this graph is $q$-regular and thus it was $q^{k + 1}$ edges, since each side of the bipartite graph has size $q^k$. Therefore, if this graph was $C_{2k}$ free, then this will be an extremal graph with this property due to the Bondy-Simonovits upper bound. Let’s look at the smallest example $k = 2$.

We have $(p_1, p_2) \sim (\ell_1, \ell_2)$ if $p_2 + \ell_2 = p_1 \ell_1$. Therefore, for any fixed $\ell = (\ell_1, \ell_2)$ the set of points adjacent to $\ell$ in $H_2(q)$ are precisely those points which lie on the line $y = \ell_1 p_1 - \ell_2$, i.e., the line of slope $\ell_1$ through the point $(0, -\ell_2)$ in the plane $\mathbb{F}_q^2$. If we identify the elements of $L$ by these non-vertical lines of $\mathbb{F}_q^2$ (and identify $P$ with the points in $\mathbb{F}_q^2$), then we get an isomorphic between $H_2(q)$ and the incidence graph between the points and non-vertical lines of $\mathbb{F}_q^2$. The vertical lines correspond to a point $P_\infty$ on the line $\ell_\infty$ that one can use to obtain the projective completion of the affine space $\mathbb{F}_q^2$. So geometrically, we can also think of $H_2(q)$ as the following graph:

Given the projective plane $\mathrm{PG}(2,q)$, let $\ell$ be a line and $P$ a point on the line $\ell$. Then $H_2(q)$ is the incidence graph between the points not lying on the line $\ell$ and the lines which do not contain the point $P$, i.e., all the lines through the $q$ points in $\ell \setminus \{P\}$

Alternatively, we can use coordinates and get the following description:

Let $P$ be the set of points with (homogeneous) coordinates $(1, p_1, p_2)$ and let $L$ be the set of  all lines through the points with coordinates $(0, 1, \ell_1)$ with $\ell_1 \in \mathbb{F}_q$. Then $H_2(q)$ is the incidence graph between $P$ and $L$

From these geometric descriptions, and the fact that through every two points there is a unique line, it is clear that $H_2(q)$ is $C_4$-free. In fact, we can give a similar geometric description of all $H_k(q)$‘s. We first give the description involving coordinates and then take a coordinate-free approach which will in fact give a broader class of graphs.

Let $P$ be the set of points in $\mathrm{PG}(k,q)$ with (homogeneous) coordinates $(1, p_1, \dots, p_k)$ and let $L$ be the set of all lines through the points with coordinates $(0, 1, \ell_1, \ell_1^2, \dots, \ell_1^{k-1})$ with $\ell_1 \in \mathbb{F}_q$. Then $H_k(q)$ is the incidence graph between $P$ and $L$

Note that the set of points we have used to define the line set is in fact a part of the normal rational curve (a.k.a. moment curve) $\{(0, \ell_1, \dots, \ell_1^{k-1}) : \ell_1 \in \mathbb{F}_q\} \cup \{(0, 0, \dots, 0, 1)\}$ in the hyperplane defined by the points whose first coordinate is $0$ (we can call it the hyperplane at infinity and the set $P$ as the affine points). The following map gives us the isomorphism between the two descriptions of the Wenger graph:  $(p_1, \dots, p_k) \mapsto (1, p_1, \dots, p_k)$  and $(\ell_1, \dots, \ell_k) \mapsto \{\lambda(1, 0, -\ell_2, -\ell_3, \dots, -\ell_k) + \mu (0, 1, \ell_1, \ell_1^2, \dots, \ell_1^{k-1}) : \lambda, \mu \in \mathbb{F}_q\}$.

The main property that the normal rational curve in $\mathrm{PG}(k - 1, q)$ has, and what we will use to show cycle freeness, is that any $k$ points on it are linearly independent. Or equivalently, any $s$-dimensional (projective) subspace of $\mathrm{PG}(k - 1, q)$ intersects the curve in at most $s + 1$ points, where $1 \leq s \leq k - 2$. The object that abstracts out this property is known as an arc, i.e., a set of points in $\mathrm{PG}(k - 1, q)$ with the property that no $k$ points on it lie on a common hyperplane. With this in our hand we get the following coordinate free description of Wenger graphs, which appears in [1] and [2]:

Construction 2: Let $H_\infty \cong \mathrm{PG}(k - 1, q)$ be a hyperlpane in $\mathrm{PG}(k, q)$ and let $S$ be an arc of size $q$ in $H_\infty$. Define $H_k(q)$ to be the incidence graph between the point set $\mathrm{PG}(k, q) \setminus H_\infty$ and the set of all lines of $\mathrm{PG}(k, q)$ that contain a point of $S$

This is in fact a larger class of graphs than the one described by Wenger since an arc does not have to come from a normal rational curve (there exist several such families of arcs). Now that we have this geometric construction, let’s focus on the graph $H_3(q)$.

Generalized Quadrangles and the Wenger Graph

Let’s see how $H_3(q)$ is related to generalized $4$-gons in exactly the same as $H_2(q)$ is related to generalized $3$-gons (the projective planes). For all $k \geq 3$, the graph $H_k(q)$ can be shown to be $C_6$-free as follows: if there was a $C_6$ then we will have $3$ lines $\ell_1, \ell_2, \ell_3$ in $\mathrm{PG}(k, q)$ which pairwise intersect each other in $\mathrm{PG}(k, q) \setminus H_\infty$ and intersect $H_\infty$ in a point of the set $S$; but this will be a contradiction to the fact that $S$ is an arc since these lines will span a plane which intersects $H_\infty$ in a line that contains $3$ points $\ell_1 \cap H_\infty, \ell_2 \cap H_\infty$ and $\ell_3 \cap H_\infty$ of $S$. Alternatively, we can do the same thing as we did before and show that $H_3(q)$ is in fact a latex $q$-regular induced subgraph of the incidence graph of a generalized quadrangle of order $q$.

The generalized quadrangle that we will use to show this is the following one, called $T_2(O)$, which was originally constructed by Tits in early 1960’s:

Let $H \cong \mathrm{PG}(2, q)$ be a hyperplane in $\mathrm{PG}(3,q)$ and let $O$ be an oval in $H$, i.e., a set of $q + 1$ points no three of which are collinear. Define the points as (i) the points of $\mathrm{PG}(3,q) \setminus H$, (ii) the hyperplanes $X$ of $\mathrm{PG}(3,q)$ for which we have $|X \cap O| = 1$, and (iii) one new symbol $(\infty)$. Define the lines as (a) lines of $\mathrm{PG}(3,q)$ through points of $O$ which are not contained in $H$ and (b) the points of $O$. A point of type (i) is only incidence with lines of type (a), and the incidence is the natural one. A point of type (ii) is incidence with all the lines of type (a) contained in it and with the unique line of type (b) corresponding to the unique element of $O$ contained in it. The point $(\infty)$ is incidence with no lines of type (a) and all lines of type (b).

Now note that if from this generalized quadrangle we remove the point $(\infty)$, a line $\ell$ of type (b) (which corresponds to a point $P$ of $O$), and all lines and points which are at distance at most $2$ from $(\infty)$ or $\ell$ in the incidence graph of $T_2(O)$, then what we are left with is the incidence structure defined on all the points of type (i) and those lines of type (ii) which pass through a point of $S = O \setminus \{P\}$This is precisely the description of the Wenger graph $H_3(q)$!

Now, one might want to find out $H_3(q)$ is related to the more well known generalized quadrangle $Q(4,q)$ (which is also the one that was constructed by Benson). This is given by the well known isomorphism between $T_2(O)$ and $Q(4,q)$ whenever $O$ is an irreducible conic (see Theorem 3.2.2 in [3]), which by the seminal work of Segre, is always true for $q$ odd.

Let’s end this blog post now with a list of questions, challenges and references.

Question 1: The graph $H_5(q)$ is known to be a homomorphic image of a $q$-regular induced subgraph of the split Cayley hexagon $H(q)$. Is there an easy way to see that?

Question 2: If we think of $H_k(q)$ as a point-line geometry, then construction 2 above gives us what is known as a linear representation of the geometry described in construction 1. So $H_3(q)$ is a linear representation of a particular subgeometry of the generalized quadrangle $T_2(O)$, and $H_k(q)$ is a generalization of that. What are some other interesting subgeometries (perhaps corresponding to regular induced subgraphs) of generalized polygons that can be generalized in this way to give interesting families of bipartite graphs that can be useful in Turán problems?

Question 3: Can $H_4(q)$ be described using some well-studied geometrical structure like the generalized polygons? May be it’s related to some near polygon or a polar space?

Big Challenge: Give a finite geometrical construction of an infinite family of $C_8$-free graphs which have $n$ vertices and $\Theta(n^{5/4})$ edges. For example, these graphs could have $\Theta(q^4)$ and $\Theta(q^5)$ edges.

[1] P. Cara, S. Rottey and G. Van de Voorde, A construction for infinite families of semisymmetric graphs revealing their full automorphism group.
[2] K. E. Mellinger and D. Mubayi, Constructions of Bipartite Graphs from Finite Geometries. Constructions of Bipartite Graphs from Finite Geometries.
[3] S. E. Payne and J. A. Thas, Finite Generalized Quadrangles.
[4] F. Lazebnik and S. Sun, Some families of graphs, hypergraphs and digraphs defined by systems of equations: a survey.
[5] S. M. Cioabă, F. Lazebnik and W. Li, On the spectrum of Wenger graphs.

The footprint bound

Studying the set of common zeros of systems of polynomial equations is a fundamental problem in algebra and geometry. In this post we will look at estimating the cardinality of the set of common zeros, when we already know that it is finite. In fact, with some extra theory one can also determine whether the set of zeros is finite or not.

My main motivation here is to popularise the so-called footprint bound, especially among combinatorialists (combinators?).  As an application, we will see a nice “conceptual” proof of the so-called Alon-Füredi theorem, which is a result that in particular solves the following well-known geometric problem whose $3$-dimensional case appeared as Problem 6 in IMO 2007:

Let $S$ be a finite subset of a field $F$ and let $a \in S^n$. What is the minimum number of hyperplanes in $F^n$ that can cover all the points in $S^n$, while missing the point $a$?

(The answer is $n(|S| - 1)$.)

To be able to state the footprint bound we need some definitions. A monomial order is a total order on the set of all monomials in $F[x_1, \dots, x_n]$ which satisfies the property $u \leq v \implies uw \leq vw$ for any monomials $u, v, w$. For example, we can look at the graded lexicographic order in which we first compare the total degree of the monomials and if that’s equal then we compare them lexicographically, i.e., for $u = (u_1, \dots, u_n)$ and $v = (v_1, \dots, v_n)$ we have $\prod x_i^{u_i} > \prod x_i^{v_i}$ if $\sum u_i > \sum v_i$ or if $\sum u_i = \sum v_i$ and the left-most non-zero entry of the vector $u - v$ is positive (so for example, we will have $x_1^2 x_2^3 \geq x_1^2 x_2 x_3^2$).

Now given a polynomial $f$ in $F[x_1, \dots, x_n]$ and a monomial ordering $\leq$, we denote the largest monomial  appearing in $f$ w.r.t. “$\leq$” by $LM(f)$ (a.k.a. leading monomial). And if $J$ is an ideal of polynomials then we define $LM(J) = \{LM(f) : f \in J\}$. Given an ideal $J$, we are going to give an upper bound on the set $V(J) = \{x \in F^n : f(x) = 0~\forall f \in J\}$. For that, let $\Delta(J)$ denote the set of those monomials which are not contained in the (monomial) ideal $\langle LM(J) \rangle$ (these are also known as standard monomials). Then this set $\Delta(J)$ is known as the footprint of the ideal $J$.

Theorem 1 (Footprint bound [7], [5,§5.3]) If $\Delta(J)$ is finite, then we have $|V(J)| \leq |\Delta(J)|$. Moreover, the bound is sharp if the ideal $J$ is radical.
Proof. Exercise! (show that $\Delta(J)$ is a basis of the vector space $F[x_1, \dots, x_n]/J$.)

Of course, the usefulness of this bound (at least in combinatorial situations) depends on how easily we can calculate, or estimate, $|\Delta(J)|$. Say $J$ is generated by $g_1, \dots, g_r$ and let $\Delta(g_1, \dots, g_r) = \{x^u : x^u \not \in \langle LM(g_i) \rangle ~\forall i\}$. Then clearly $\Delta(J) \subseteq \Delta(g_1, \dots, g_r)$, and since we have a finite list of monomials to check, with right choice of $g_i$‘s it can be easier to compute $\Delta(g_1, \dots, g_r)$. Ideally, we would like $\Delta(J)$ to be equal to $\Delta(g_1, \dots, g_r)$, and in fact this does happen when $g_1, \dots, g_r$ forms a Gröbner basis of $J$ (if you know what a Gröbner basis is; but you don’t really need to for the rest of this post). Let’s consider an example. Let $n = 2$ and $J = \langle x^2 y^3 - y, x^3y - x, x^4 - y^3, y^4 - xy^2 \rangle \subset F[x, y]$, and consider the graded lexicographical order. Then the multiples of the leading monomials $x^2y^3, x^3y, x^4, y^4$ can be seen pictorially as follows:

The monomials which are multiples of these four monomials, i.e. the monomials in the ideal generated by these leading monomials, correspond to the integer points in the shaded region. Therefore, the elements of $\Delta(x^2 y^3 - y, x^3y - x, x^4 - y^3, y^4 - xy^2)$ corresponds to the points in the unshaded region, and hence $|\Delta(J)| \leq |\Delta(x^2 y^3 - y, x^3y - x, x^4 - y^3, y^4 - xy^2)| = 12$, which implies that these polynomials have at most $12$ common solutions (Wolfram Alpha tells me that there are in fact only $2$).

Say we are now interested in estimating the number of zeros of a polynomial $f \in F[x_1, \dots, x_n]$ contained in a finite grid $S = S_1 \times \cdots \times S_n$, where $S_i \subseteq F$, assuming that $f$ does not vanish on all points of $S$. This is precisely what the Alon-Füredi theorem is all about. Let $s_i = |S_i|$ and $g_i = \prod_{a \in S} (x_i - a)$. Then $g_i$ is a polynomial of degree $s_i$, and the points of the $S$ are the common solutions of the polynomials $g_1, \dots, g_n$. What we are interested in is $V(J)$ for $J = \langle f, g_1, \dots, g_n \rangle$. For that we will first assume that $f$ is in its reduced form, so that $LM(f)$ (with respect to any monomial ordering) is of the form $\prod x_i^{u_i}$ where $0 \leq u_i \leq s_i - 1$ for all $i$. We can always do this by reducing higher powers of $x_i$ by using the equation $g_i(x_i) = 0$ (the so-called grid reduction [3, Chapter 8]). Say $\deg (f) = d$ and $x^u = \prod x_i^{u_i}$ is the leading monomial of $f$, with $\sum u_i = d$. Then $\Delta(x^u, x_1^{s_i}, \dots, x_n^{s_n})$ is equal to the set of all monomials of the form $\prod x_i^{v_i}$ where there exists an index $j$ for which $v_j < u_j$. The number of such monomials is in fact equal to $\prod s_i - \prod (s_i - u_i)$ because $\prod s_i$ is the total number of reduced monomials, and $\prod (s_i - u_i)$ of them are those which are multiples of $x^u$. If we let $c_i = s_i - u_i$, then we have $1 \leq c_i \leq s_i$ and $\sum c_i = (\sum s_i) - d$, which directly implies the Alon-Füredi theorem:

Theorem 2 (Alon-Füredi [1, Theorem 5]) Let $f$ be a polynomial which does not vanish on the entire grid $S = S_1 \times \cdots \times S_n$, then $f$ does not vanish on at least $\min \{ \prod c_i : 1 \leq c_i \leq s_i ~\forall i,~ \sum c_i = (\sum |S_i|) - \deg f\}$ points of $S$.

Compared to the other proof of the Alon-Füredi theorem, I feel like this proof gives us a better understanding of where that bound comes from and “why” this theorem is true. Also, the same proof also gives the generalized version of the Alon-Füredi theorem [3, Theorem 9.1.2] (over fields). Moreover, one can also prove Alon’s combinatorial nullstellensatz using the footprint bound as follows.

Theorem 3 (Combinatorial Nullstellensatz) Let $f$ be such that $LM(x) = x^u$  with $u = (u_1, \dots, u_n)$ satisfying $u_i < |S_i|$ for all $i$. Then there exists an $a \in S$ for which $f(a) \neq 0$.
Proof. The size of the footprint of the ideal $I = \langle f, g_1, \dots, g_n \rangle$ is strictly less than $|S|$ since $u$, being the exponent of a leading monomial, is not contained in it. Therefore by the footprint bound we have $|V(I)| < |S|$, which implies that there is a point where $f$ does not vanish.

So, footprint bound is a nice algebraic result which has several interesting applications and extensions [2, 4, 6, 7, 8, 9, 10]. It is particularly useful for studying systems of polynomial equations over a finite field $\mathbb{F}_q$, where we can take the grid $S$ to be equal to $\mathbb{F}_q^n$. For further reading on this bound see the papers in the references below.

And now for the extra theory, here is a characterisation of systems of polynomial equations which have only finitely many common zeros (the so-called zero dimensional affine varieties over the algebraic closure of the field).

Theorem 4 ([5, Page 234]) Let $k$ be an algebraically closed field and let $V = V(I)$ be an affine variety in $k^n$ where $I$ is an ideal in $k[x_1, \dots, x_n]$. Then the following statements are equivalent:

1. $V$ is a finite set.
2. For each $i$, there exists some $m_i \geq 0$ such that $x_i^{m_i} \in \langle LM(I) \rangle$.
3. Let $G$ be a Gröbner basis for $I$. Then for each $i$, there exists some $m_i \geq 0$ such that $x_i^{m_i} = LM(g_i)$ for some $g_i \in G$.
4. The set $\Delta(I)$ is finite.
5. The $k$-vector space $k[x_1, \dots, x_n]/I$ is finite-dimensional.

References

[1] N. Alon and Z. Füredi. Covering the cube by affine hyperplanes. European J. Combin., 14(2):79–83, 1993.
[2] P. Beelen, M. Datta, and S. R. Ghorpade. Vanishing ideals of projective spaces over finite fields and a projective footprint bound. arXiv:1801.09139.
[3] A. Bishnoi, Some contributions to incidence geometry and polynomial method, PhD Thesis (2016), Ghent University.
[4] C. Carvalho. On the second Hamming weight of some Reed–Muller type codes. Finite Fields Appl., 24:88–94, 2013.
[5] D. Cox, J. Little, and D. O’Shea. Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2007.
[6] O. Geil. On the second weight of generalized Reed-Muller codes. Designs, Codes and Cryptography, 48(3): 323-330, 2008.
[7] O. Geil and T. Høholdt. Footprints or generalized Bezout’s theorem. IEEE Trans- actions on Information Theory, 46(2):635–641, 2000.
[8] O. Geil and C. Thomsen. Weighted Reed–Muller codes revisited. Designs, Codes and Cryptography, 66(1):195–220, 2013.
[9] O. Geil and U. Martínez-Penãs. Bounding the number of common zeros of multivariate polynomials and their consecutive derivatives, arXiv:1707.01354.
[10] R. Pellikaan and X.-W. Wu. List decoding of q-ary Reed-Muller codes. IEEE Transactions on Information Theory, 50(4):679–682, 2004.

Introduction to polynomial method

(The following is a blog-friendly version of Chapter 7 of my PhD thesis, which is an introduction to the so-called polynomial method.)

The polynomial method is an umbrella term for different techniques involving polynomials which have been used to solve several problems in finite geometry, discrete geometry, extremal combinatorics and additive number theory. One of the general philosophies behind this method is to associate a set of polynomials (possibly a single polynomial), to a combinatorial object that we want to study, and then use some properties of these polynomials to describe the combinatorial object. For a concrete example, let us go through Koornwinder’s proof of the absolute bound on equiangular lines.

A set of lines in the Euclidean space $\mathbb{R}^n$ through the origin (or any other fixed point) is called equiangular if the angle between every pair of distinct lines in the set is the same. For example, joining the opposite vertices of a regular hexagon in the plane $\mathbb{R}^2$, we get three equiangular lines.

At most how many equiangular lines can there be in $\mathbb{R}^n$?

This question was addressed by Gerzon (as reported by Lemmens and Seidel) who proved that there are at most $n+1$ equiangular lines in $\mathbb{R}^n$. Thus in particular, the regular hexagon example gives us the maximum possible equiangular lines in $\mathbb{R}^2$. But in general this bound is not sharp. In 1976, Koornwinder gave an “almost trivial proof” of Gerzon’s bound by giving a bijective correspondence between the set of equiangular lines in $\mathbb{R}^n$ and a linearly independent set of polynomials lying in an ${n+1 \choose 2}$ dimensional vector  space. This correspondence is as follows. Let $L_1, \dots , L_k$ be $k$ equiangular lines in $\mathbb{R}^n$ and let $u_1, \dots, u_k$ be unit vectors on these lines, chosen arbitrarily. Then we have $\langle u_i,u_j \rangle^2 = \alpha$ for $i \neq j$ where $\alpha$ is a fixed real number in the interval $[0,1)$. For $i \in \{1, \dots, k\}$ define $f_i \in \mathbb{R}[t_1, \dots, t_n]$ by $f_i(t_1, \dots, t_n) = \langle u_i, (t_1, \dots, t_n) \rangle^2 - \alpha^2 (t_1^2 + \dots + t_n^2)$. Now since $f_i(u_j) = 0$ for all $i \neq j$ and $f_i(u_i) = 1 -\alpha^2 \neq 0$, it is easy to see that $f_1, \dots, f_k$ are linearly independent polynomials in the vector space $\mathbb{R}[t_1, \dots, t_n]$. As these polynomials lie in the ${n + 1 \choose 2}$ dimensional subspace spanned by the monomial set $\{t_i t_j : 1 \leq i < j \leq k\}$, we get $k \leq \binom{n + 1}{2}$.

The argument above can also be seen as an example of the linear algebra method in combinatorics, which has been discussed in much detail in the influential (unfinished) manuscript of Babai and Frankl, and more recently in the beautiful book by Matoušek.

Another important way of using polynomials is to capture the combinatorial object via zeros of polynomials (or in general, algebraic varieties). One of the earliest examples here is the determination of the minimum size of an affine blocking set by Jamison in 1977. The problem is to find the minimum number of points required to “block” every hyperplane of the affine space $\mathbb{F}_q^n$. Clearly $n(q - 1) + 1$ points suffice (by taking all points that lie on one of the $n$ axes), but can we do better? Jamison proved that we cannot, and his polynomial method proof can be sketched as follows: (a) first consider the dual problem, which is equivalent to finding the minimum number of hyperplanes required to cover all points of $\mathbb{F}_q^n$ except the origin, (b) then identify $\mathbb{F}_q^n$ with the finite field $\mathbb{F}_{q^n}$, (c) finally associate each hyperplane with the minimal polynomial over $\mathbb{F}_q^n$ that vanishes on the hyperplane to show (using the theory of linearized polynomials) that if the number of hyperplanes is less than $n(q - 1) + 1$, then the polynomial $t^{q^n - 1} - 1 = \prod_{\alpha \in \mathbb{F}_q^\times} (t - \alpha)$ does not divide the product of these polynomials corresponding to the hyperplanes. This technique of using polynomials over finite fields to solve finite geometrical problems came to be known as the “Jamison method” and it saw several applications (see for example this and this).

Brouwer and Schrijver gave another proof of Jamison’s theorem where they also started by considering the dual problem of hyperplane coverings but then proceeded by a much simpler argument involving multivariate polynomials over finite fields. Their approach was in fact quite similar to Chevalley’s proof of the famous Chevalley-Warning theorem  using reduced polynomials. We will see in Chapters 8 and 9 how both of these results are linked together by the notion of grid reduction, and in particular by the Lemma that a polynomial $\mathbb{F}_q[t_1, \dots,t_n]$ which vanishes on all points of $\mathbb{F}_q^n$ except one must have degree at least $n(q-1)$. The Chevalley-Warning theorem, which is a statement on the zero set of a collection of polynomials over a finite field, has also found several applications in combinatorics. Alon, Friedland and Kalai used it to prove that every 4- regular graph plus an edge contains a 3-regular subgraph. Later, Bailey and Richter used the Chevalley-Warning theorem to give a new proof of the famous Erdős-Ginzburg- Ziv theorem from additive number theory. Recently, Clark, Forrow and Schmitt have shown that the Chevalley-Warning theorem and its combinatorial applications can be derived, and further generalized, using a result of Alon and Füredi from 1993. We will devote Chapter 9 to this Alon-Füredi Theorem, where we generalize the result and give a new simple proof. We also show how this result is linked to several other topics like coding theory, finite geometry and polynomial identity testing.

An important tool in the polynomial method involving zeros of polynomials is a result called Combinatorial Nullstellensatz. This powerful tool and its generalizations have been used extensively to solve several problems in additive number theory (see Chapter 9 of this for a survey) and more recently in some other areas as well. In 2014, Clark revisited Alon’s Combinatorial Nullstellensatz and showed how its proof can be seen as a “restricted variable analogue” of Chevalley’s proof of the Chevalley-Warning Theorem. He further generalized this result to commutative rings (adding certain extra conditions) and made it clear how many of the ideas involved are related to the notion of grid reduction. Ball and Serra introduced a related result which they called Punctured Combinatorial Nullstellensatz. This result was proved using Alon’s Combinatorial Nullstellensatz, and it has several combinatorial applications of its own. We will give another proof of this result in Chapter 8 by directly using the notion of grid reduction, and then use this result to prove a new generalization of the Chevalley-Warning theorem which we call the Punctured Chevalley-Warning Theorem. In fact, this result generalizes Brink’s Restricted Variable Chevalley-Warning theorem.

In recent years, there has been a lot of interest in the polynomial method as a result of Dvir’s two-page proof of the finite field Kakeya problem which involved an easy polynomial argument, and the developments which followed. Many experts worked on the finite field Kakeya problem using different techniques involving algebraic geometry and Fourier analysis, but made only partial progress towards a solution to this problem. And thus it was a great surprise to the mathematical community that such an easy polynomial argument could resolve this famous open problem. Even more recently, the notorious cap set problem has been solved using the polynomial method by Ellenberg and Gijswijt. Ideas originating from Dvir’s work have lead to several important advancements in mathematics, including the big breakthrough in the famous Erdős distinct distances problem by Guth and Katz. It is interesting to note that Dvir’s polynomial technique is a bit different from the techniques we have mentioned so far in this introduction as it involved polynomial interpolation instead of constructing explicit polynomials. For more details on this, we recommend the surveys by Dvir and Tao, and the recent book by Guth. Another example where a combinatorial problem is solved using polynomial interpolation, combined with a geometric argument, is Segre’s classical theorem on ovals in finite projective planes. Interestingly, the so-called “lemma of tangents” of Segre was used in combination with the Jamison/Brouwer-Schrijver bound on affine blocking sets by Blokhuis and Korchmáros to solve the Kakeya problem in $2$ dimensions. Segre’s result (and his lemma of tangents) has been generalized further to higher dimensional finite projective spaces by Ball. For more on polynomial method in finite geometry, see the survey by Ball.

On a famous pigeonhole problem

After a short break from blogging, which involved moving from Ghent to Berlin, dealing with German bureaucracy, and learning how to make simple websites (the easiest bit), I am now back. I am working as a postdoc at the Free University of Berlin right now and a part of my job is to teach a course (mainly taking care of the exercise classes) called Extremal Combinatorics with Tibor Szabó . While preparing a review sheet of exercises for this course, we discussed about which pigeonhole problem to include. In the actual sheet we ended up giving a simple one, but we did think about including the following famous problem:

Prove that if you pick any $n + 1$ numbers from $1, 2, \dots, 2n$, there will be two distinct numbers $a$ and $b$ such that $a$ divides $b$.

This used to be one of the favourite problems of Paul Erdős for young (mathematically inclined) kids. Quoting Proofs from THE BOOK, “As Erdős told us, he put this question to young Lajos Pośa during dinner, and when the meal was over, Lajos had the answer. It has remained Erdős’ famous initiation questions to mathematics”. Perhaps you should try it over your next dinner, before reading further.

So here is the standard pigeonhole argument for this. Every positive integer $a$ can be written as $a = 2^k(2m - 1)$, with $k \geq 0$ and $m \geq 1$ by considering the largest power of $2$ in the factorisation of $a$. Now write each of the $n + 1$ numbers you have chosen in this form. Since there are precisely $n$ odd integers in the set $1, \dots, 2n$ and hence precisely $n$ possibilities for the $2m - 1$ factor, by the pigeonhole principle we must have two integers $a, b$ with $a = 2^{k_1}(2m - 1)$ and $b = 2^{k_2}(2m - 1)$, for some $k_1 < k_2$ and $m \in \{1, \dots, n\}$. Thus, $a$ divides $b$.

Looking at the proof carefully, what we have really done is partition the set $[2n] = \{1, \dots, 2n\}$ into $n$ parts $S_1, \dots, S_n$, with $S_i = \{2^k(2i - 1) : k \geq 0\} \cap [2n]$, such that in each $S_i$ every number divides all the other numbers which are bigger. Then since we have chosen $n + 1$ elements from $[2n] = S_1 \sqcup \dots \sqcup S_n$, there must be an index $i$ for which $S_i$ contains two of these $n + 1$ elements.

Now, as a mathematician, there are two natural questions that one can ask:

(Q1) What is the largest possible size of a subset of $[2n]$ which does not contain two elements with one dividing the other?

(Q2) Can we classify, or say something interesting about, all such largest possible subsets?

(Q1) is straightforward to answer, since we quickly see that the $n$-element subset $\{n + 1, n + 2, \dots, 2n\}$ does not have any two distinct elements in it with one dividing the other. And since we have shown that any $(n+1)$-element subset will violate this property, $n$ is the largest possible size.

(Q2) is what we will be really interested in.

This is such a natural question for a combinatorialist, but somehow I have been unable to find any reference for it (which Tibor found quite surprising as well). Now the first natural guess for an answer is that the example $\{n + 1, n + 2, \dots, 2n\}$ is the unique $n$-element subset of $[2n]$ with this “non-divisibility” property. It is natural because this is what happens in typical extremal combinatorics problems. But if you think about it for a while, you’ll be able to come up with the set $\{n, n + 1, n + 2, \dots, 2n - 1\}$, which is another such example. And then $\{n - 1, n, n + 1, \dots, 2n - 3, 2n - 1\}$ is another one. We can keep doing this for a certain $k = cn$ number of steps, where $c$ is a constant (figure out what $c$ is!). This shows that there are at least a linear number of extremal examples in this problem. Moreover, you can prove that any such extremal set must contain all the odd numbers from $\{n + 1, n + 2, \dots, 2n\}$. But are there any more such sets?

After some more thought, I was able to show that there are in fact exponentially many such extremal sets! More precisely, there exists a constant $c > 1$ such that there are at least $c^n$ $n$-element subsets of $[2n]$ in which there are no two elements with one dividing the other. I will leave the proof of this as an exercise, since it did not take me too long to come up with it and in fact, one of the students in the exercise classes that I teach was able to come up with the same proof as mine pretty quickly.

Now the questions that I want to ask are the following:

What is the best possible value of the base $c$? In other words, asymptotically how many such extremal subsets can we find? Is there a “substantially” better upper bound on the number of such sets than $\binom{3n/2}{n/2} \sim 2^{1.38 n}$?

Edit 06/11/2017: I computed the number of such extremal subsets for small values of $n$, after which a quick search on The On-Line Encyclopedia of Integer Sequences revealed that people have computed and studied this number.  In particular, Robert Israel mentions the same lower bound that I obtained.

Edit 22/05/2018: My question has been completely answered independently by Liu, Pach and Palincza in “The Number of Maximum Primitive Sets of Integers“, Edit 29/11/2018: and Nathan Mcnew in “Counting primitive subsets and other statistics of the divisor graph of {1, …, n}”

Posted in Combinatorics, Extremal Combinatorics | | 5 Comments

What I have learned in finite geometry

On September 2nd, 2014 I wrote a blog post titled learning finite geometry, in which I described how much I have learned in my first year of PhD and more importantly, the topics that I wish to learn while I am in Ghent. Since I’ll be leaving Ghent in a few days, I feel like this is the right time reflect on that old post. I am quite happy to see that beyond just learning those topics, I have also been able to do some research on them. Here are the topics that I described in my previous blog post:

1. Blocking sets
Two of my papers have results on blocking sets. In the first one, I used the Alon-Furedi theorem to prove old and new results on partial covers and blocking sets with respect to hyperplanes in finite Desarguesian affine/projective spaces. I have given a new common framework for treating some of the problems on blocking sets in an easy way using the polynomial method. See section 6 of my paper for the details.

In my second paper, I have proved a generalisation of a classical result in finite geometry from the 1970s due to Aiden A. Bruen and Jeff Thas. Moreover, I have done so with a new method involving the expander mixing lemma which has lead to a unified (and simple) approach to problems related to blocking sets, Nikodym sets and tangency sets. This method is quite interesting in its own right as it has also lead to some advancements in the cage problem.

2. Unitals
These objects appear in my second paper on blocking sets, as they were used by Sam to give a construction of large minimal multiple blocking sets in finite projective planes. In fact, as Sam has described in his blog post, an open problem on unitals was one of my motivations to look at these blocking sets. Sadly, we haven’t been able to solve that open problem yet.

3. Polar Spaces
After the end of my first year, I had seen polar spaces appear in almost every other research paper and talk in finite geometry. Even though I had read and understood the definitions of these objects in my first year, I wasn’t really comfortable with them. I am not sure when exactly that changed, but it was certainly quite gradual. In my paper with John and Gordon, on regular induced subgraphs of generalized polygons, I have used some of the basic theory of polar spaces to give new constructions of small $k$-regular graphs of girth $g$ with $g = 8, 12$. Before that I also refereed a paper which helped me revisit some of the basics of the axiomatic theory of polar spaces. Overall, I can say that I have made pretty good progress towards understanding polar spaces. I now wish that more mathematicians would learn the basic theory of these beautiful geometrical objects, especially because people sometimes obtain results on specific classes of finite polar spaces without realising it (see this for example). The new book “Finite Geometry and Combinatorial Applications” by Simeon Ball can be a useful resource for anyone interested in doing so.

Besides these topics, I have also learned about several combinatorial problems in which finite geometry plays an important part. For example, the cage problem, forbidden subgraphs, and ramsey numbers. During my postdoc years, I will work on finding more such combinatorial applications of finite geometry, and using various tools from combinatorics to solve problems in finite geometry.

The cage problem and generalized polygons (part 1)

This post is a continuation of my previous post on the cage problem. Just to recall the main problem, for any given integers $k \geq 2$ and $g \geq 3$, we want to find the least number of vertices in a simple undirected graph which is $k$-regular and has girth $g$. The least number is denoted by $c(k, g)$ and the $(k, g)$-graphs with $c(k, g)$ vertices are called cages. As we saw before, besides the generalized polygons of order $q$, for some prime power $q$, there are no know infinite families of cages (for a list of all known cages, see the survey by Excoo and Jajcay. ). And by the result of Feit and Higman on generalized polygons, we see that these infinite families have $g \in \{6, 8, 12\}$. It turns out that we know a lot more about $c(k, g)$ for $g \in \{6, 8, 12\}$ than the general case, even when there is no known generalized polygon of order $k - 1$, i.e., when $k - 1$ is not a prime power.

Let’s start with an example. For every prime power $q$, we know that $c(q + 1, 6) = 2(q^2 + q + 1)$ by looking at the incidence graph of a finite projective plane (i.e., a generalized $3$-gon) $\pi$ of order $q$. Now pick a point $X$ in $\pi$ and a line $\ell$ through $X$. Then observe that through each point $Y \neq X$, there are exactly $q$ lines which do not contain $X$: all the lines through $Y$ except the line $XY$. Dually, every line $m \neq \ell$ contains exactly $q$ points which do not lie on $\ell$: all points on $m$ except the intersection point of $\ell$ and $m$. What this shows is that if we take the subgraph of the incidence graph of $\pi$ induced on the $q^2$ points not contained in $\ell$ and the $q^2$ lines not containing $X$, then it is a $q$-regular graph. Moreover, this graph has girth $6$, which leave as an exercise to the reader. So, we have just constructed a $q$-regular graph of girth $6$ with $2q^2$ vertices, proving that $\displaystyle c(q, 6) \leq 2q^2$ for all prime power $q$.

In fact, we can do slightly better if we take $X \not\in \ell$. We then take all the points distinct from $X$ which are not on $\ell$, and all the lines distinct from $\ell$ which do not contain $X$. This gives us a $q$-regular subgraph on $2(q^2 - 1)$ vertices, proving $c(q, 6) \leq 2(q^2 - 1)$ (not a big improvement, I know). But is this the best that we can do by pursuing this idea of constructing $q$-regular induced subgraphs?

Interestingly, there is something much better that can be done, if $q$ is a square. We can take the subgraph induced on the points and lines of $\pi$ which are not contained in a Baer subplane of $\pi$. In the Desarguesian plane $\mathrm{PG}(2,q)$, we can get a Baer subplane by restricting the co-ordinates of the points and lines to the subfield $\mathbb{F}_{\sqrt{q}}$ (similarly the real projective plane $\mathrm{PG}(2, \mathbb{R})$ is a Baer subplane of the complex projective plane $\mathrm{PG}(2, \mathbb{C})$). While every subfield of $\mathbb{F}_q$ gives rise to a subplane of $\mathrm{PG}(2,q)$, the Baer subplane in particular has the following nice properties:

• through every point outside the subplane there is a unique line of the subplane;
• every line outside the subplane contains a unique point of the subplane;

In other words, the induced subgraph that we get on the points and lines not contained in the Baer subplane is $q$-regular. This construction gives us $c(q, 6) \leq 2(q^2 + q + 1 - q - \sqrt{q} - 1) = 2(q^2 - \sqrt{q})$, which is much better than the previous bound of $2(q^2 - 1)$. But, this construction only works when $q$ is a square.

So now the natural question arises (as it always should!), is this the best that we can do? Interestingly, now the answer is yes. We have actually reached a theoretical lower bound on the size of a $q$-regular induced subgraph in a projective plane of order $q$. This is a consequence of the following result that tells us how small a $k$-regular induced subgraph of a $d$-regular bipartite graph be.

Theorem 1. Let $G = (L, R, E)$ be a $d$-regular bipartite graph and let $\lambda$ be its second largest eigenvalue. Let $H$ be a $k$-regular induced subgraph of $G$. Then $|V(H)| \geq (k - \lambda)|V(G)|/(d + \lambda)$.

Theorem 1 can be proved easily using the expander mixing lemma, which I discussed in a previous post. It is well known, and easy to show, that the second largest eigenvalue of the incidence graph of a generalized $n$-gon of order $q$ is $\sqrt{q}, \sqrt{2q}$ and $\sqrt{3q}$ for $n = 3, 4$ and $6$, respectively. The number of vertices in the incidence graphs of a generalized $n$-gon is $q^2 + q + 1, (q + 1)(q^2 + 1)$ and $(q + 1)(q^4 + q^2 + 1)$ for $n = 3, 4$ and $6$ respectively. If we now substitute these values in Theorem 1, and note the factorisations $q^2 + q + 1 = (q + \sqrt{q} + 1)(q - \sqrt{q} + 1)$, $q^2 + 1 = (q + \sqrt{2q} + 1)(q - \sqrt{2q} + 1)$ and $q^4 + q^2 + 1 = (q^2 + q + 1)(q + \sqrt{3q} + 1)(q - \sqrt{3q} + 1)$ (thanks Sam!), we get the following result.

Theorem 2. Let $G$ be the incidence graph of a generalized $n$-gon of order $q$, and let $H$ be a $(q + 1 - t)$-regular induced subgraph, for some $t \geq 1$. Then

$\displaystyle|V(G)| - |V(H)| \geq \begin{cases} 2t(q + \sqrt{q} + 1), & \text{if } n = 3;\\ 2t(q + 1)(q + \sqrt{2q} + 1), & \text{if } n = 4; \\ 2t(q+1)(q^2 + q + 1)(q + \sqrt{3q} + 1), & \text{if } n = 6. \end{cases}$

In particular, for $n = 3$ and $t = 1$, we get $2(q^2 + q + 1) - |V(H)| \geq 2(q + \sqrt{q} + 1)$, i.e., $V(H) \leq 2(q^2 - \sqrt{q})$. Therefore, the construction using the Baer subplane is the best possible. In fact, for $n = 3$ and arbitrary $t$, we have a construction using $t$ disjoint Baer subplanes (see Construction 3.7 here) which is sharp according to this new bound. So, the problem of looking at regular induced subgraphs is settled for projective planes. For generalized quadrangles and hexagons though, there is a lot more to be said. In collaboration with John Bamberg and Gordon Royle, I have obtained some new finite geometrical constructions which improve the bounds on $c(q, 8)$ and $c(q, 12)$. We will discuss that in the next post.