Extremal combinatorics in generalized Kneser graphs

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1 Extremal combinatorics in generalized Kneser graphs Mussche, T.J.J. DOI: 0.600/IR Published: 0/0/2009 Document Version Publisher s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. The final author version and the galley proof are versions of the publication after peer review. The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication Citation for published version (APA): Mussche, T. J. J. (2009). Extremal combinatorics in generalized Kneser graphs Eindhoven: Technische Universiteit Eindhoven DOI: 0.600/IR General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal reuirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal? Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 25. Dec. 208

2 Extremal combinatorics in generalized Kneser graphs PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op donderdag 6 april 2009 om 6.00 uur door Tim Joris Jacueline Mussche geboren te Eeklo, België

3 Dit proefschrift is goedgekeurd door de promotoren: prof.dr. A.E. Brouwer en prof.dr. A.M. Cohen Copromotor: dr. A. Blokhuis

4 Contents Contents 3 Introduction 7. Sets Graphs Basic definitions Graph colorings and chromatic numbers Graph homomorphisms Finite projective spaces Finite fields Projective spaces The projective space PG(n, ) Some counting in PG(n, ) Finite polar spaces Combinatorics and -analogues An easy example Sperner s Theorem Original problem analogue Erdős-Ko-Rado theorem Original problem Analogue Bollobás s theorem Set version Set vs. subspace version Subspace version The Hilton-Milner theorem Original problem

5 4 CONTENTS The -analogue Small maximal cliues The Kneser and -Kneser graphs Definitions and properties Homomorphisms Homomorphisms between Kneser graphs Homomorphisms between -Kneser graphs A homomorphism from a -Kneser graph into a Kneser graph Chromatic numbers The Kneser graphs The -Kneser graphs A family of point-hyperplane graphs Definition PH(2, ) PH(3, ) Polar versions of the -Kneser graphs Definition Chromatic numbers K Q+ (2m + 2, m + ), m 2 even, a trivial case K P (n, ) Γ P (n, 2) and K P (n, 2), where P has rank Generalized Kneser graphs Chevalley groups Tits systems Coxeter systems Root systems Chevalley groups Generalized Kneser graphs Definition Parameters Chromatic numbers in the thin A n () case Chromatic numbers in the thick A n case

6 CONTENTS 5 A Parameters of generalized Kneser graphs 23 A. The case J w 0 = J A.2 The case J w0 J Bibliography 48 Index 53 Samenvatting 57 Summary 59 Acknowledgements 6

7 6 CONTENTS

8 Chapter Introduction In this first chapter we will introduce the basic notions used in this thesis. It is not the intention to give a complete, self contained description of every topic. Instead the most important notions are defined and the most useful results are mentioned, often without proof. We refer the reader to the citations with each topic for more background information and proofs of the results. First we introduce some notions of set theory. Then some graph theoretic topics are mentioned. Since this thesis describes a way of translating problems in set theory and graph theory into finite geometry we need to introduce the basic facts about projective spaces. In Chapter 5 we generalize the construction over projective spaces of Chapter 3 to work over polar spaces, so we define here what polar spaces are and give some properties. Unless stated otherwise all the objects used here are finite.. Sets Take a set X with n elements. Such a set is also called an n-set. We can ask ourselves how many subsets of X with k elements (k-subsets) there are, where 0 k n. This is of course a very easy calculation, but in a later section we will use the same method to calculate some numbers about vector spaces, so we will give the calculation anyway. To form such a k-subset we have to choose k elements from X. For the first element we have n choices, for the second element n choices, and so on. Finally for the k-th element 7

9 8 CHAPTER. INTRODUCTION we have n k + choices left. But now there are many choices that lead to the same k-subset; indeed all choices of the same k points but in a different order give rise to the same subset. So, to get the correct number we have to divide by the number of permutations of k points, which is k!. So we have: (the number of k-subsets of an n-set) = = = n(n )(n 2) (n k + ) k! n! k!(n k)! ( ) n k Definition.. The number of k-subsets of an n-set (with k n) is given by the binomial coefficient ( n k). Definition.2. A set of sets F is called a set system or a family of sets. If all the elements of F are subsets of some set X, then F is called a family over X, and X is called the universe of F. If all the members of F have size k, then F is called a k-uniform family. F is called uniform if it is k-uniform for some integer k. We give two examples of special set systems. Definition.3. A set system F is called a chain when for any two elements one is a proper subset of the other. Definition.4. A family of sets F is called an antichain if no set in F is a proper subset of another set in F. Easy examples of such families are uniform families. Antichains are also named Sperner families after E. Sperner, who proved a tight upper bound on the size of an antichain (see Theorem 2.2). A set system can be viewed as a poset (or partially ordered set). To define this notion we need to define a partial order.

10 .2. GRAPHS 9 Definition.5. A relation on a set X is called a partial order on X if it satisfies: Reflexivity: x x for all x X, Antisymmetry: if x y and y x then x = y for all x, y X, and Transitivity: if x y and y z then x z for all x, y, z X. If x y or y x then x and y are called comparable with respect to, otherwise they are called incomparable. Definition.6. A poset (or partially ordered set) is an ordered pair (X, ) where X is a set, called the ground set, and a partial order on X. A chain in a poset is a subset of X whose elements are pairwise comparable. An antichain is a subset whose elements are pairwise incomparable. A maximal chain is a chain that cannot be extended to a larger chain. It is clear to see that the relation ( is subset of ) is a partial order. So the pair (F, ), where F is a set system is a poset. It is easy to see that a(n) (anti-)chain in this poset is exactly what we called a(n) (anti-)chain before. The poset (2 X, ), where 2 X is the set system containing all subsets of X, is called the subset poset of X..2 Graphs.2. Basic definitions Definition.7. A (simple) graph Γ = (V, E) is a pair consisting of a set V of vertices and a 2-uniform family E, called the edge set, with universe V, whose members are called edges. Two vertices u, v V are called adjacent if {u, v} E and this edge is said to join the vertices u and v. In that case, u and v are also called the end points of the edge {u, v}. The neighbors of a vertex are the vertices adjacent to that vertex. The degree of a vertex is the number of its neighbors. A graph in which each vertex has the same degree is called regular. We will sometimes use Γ as a notation for the number of vertices of Γ.

11 0 CHAPTER. INTRODUCTION Definition.8. Two vertices u, v V are called joined if there are vertices u = u 0, u,..., u m = v in V for a certain m 0 such that {u i, u i+ } is an edge for all 0 i m. Such a seuence of adjacent vertices is called a path between u and v. If all the u i s are different, this is called a simple path. If u and v are connected, the length of the shortest path between them is called the distance and is denoted by d(u, v). A path of length at least three without repeated vertices except that the two endpoints are the same is called a cycle. Being joined by a path is an euivalence relation on the set of vertices. The euivalence classes are called the connected components of the graph. A graph is called connected when it has precisely one connected component. Thus the empty graph (the graph with no vertices) is disconnected because it has no connected components. Some subsets of the vertex set have special properties: Definition.9. An independent set is a subset of the vertex set in which no two vertices are adjacent. A cliue is a subset of the vertex set in which all pairs of vertices are adjacent. A complete graph is a graph where the edge set E contains all pairs of vertices in V. This means that all vertices are pairwise adjacent. The complete graph on n vertices is denoted by K n. Starting from one graph, there are many ways to construct other graphs. The most straightforward ways are taking subgraphs and taking the complement: Definition.0. A graph Γ = (V, E ) is called a subgraph of a graph Γ = (V, E) if V is a subset of V and all edges in E are also in E. A subgraph is called induced if all unordered vertex pairs of Γ that are adjacent in Γ are also adjacent in Γ. Definition.. The complement of a graph Γ = (V, E) is the graph Γ = (V, E) where the edge set E consists of all pairs of vertices that are not elements of E. In other words, adjacent pairs in Γ are not adjacent in Γ and vice versa. It is clear that the complement of a cliue is an independent set and that the complement of an independent set is a cliue. For that reason, an independent set is also called a cocliue. The complement of the complete

12 .2. GRAPHS graph K n has n vertices and no edges and is called an edgeless graph. To each graph we can associate certain numbers, called graph parameters. Some well-known examples are the independence number and the cliue number. Definition.2. The independence number α(γ) of a graph Γ is the size of the largest independent set in Γ. The cliue number ω(γ) is the size of the largest cliue in Γ. Again it is clear that α(γ) = ω(γ) and ω(γ) = α(γ)..2.2 Graph colorings and chromatic numbers Definition.3 (Graph coloring). A vertex coloring of a graph Γ = (V, E) is a partition of the vertex set. The color of a vertex is determined by the block of the partition that the vertex is in. A proper (vertex) coloring is a vertex coloring such that the end points of an edge have a different color. From now on, if the term (graph) coloring is used without further ualification, we are referring to a proper vertex coloring of a graph. Colors will usually be denoted by integers. We can now define some other graph parameters: Definition.4. The chromatic number χ(γ) of a graph Γ is the minimum number of colors needed to color Γ. A coloring using only χ(γ) colors is called a minimal coloring of Γ. Because all vertices of the same color form an independent set, there is a very easy connection between the independence number and the chromatic number of a graph: Proposition.5. For each graph Γ we have that Γ α(γ)χ(γ). Graph colorings are generalized by multiple graph colorings:

13 2 CHAPTER. INTRODUCTION Definition.6 (Multiple coloring). A k-fold coloring of a graph Γ for a positive integer k is an assignment of exactly k colors to each vertex of Γ such that two adjacent vertices have no colors in common. A connected graph Γ with at least two vertices must have at least one edge. In a k-fold coloring of Γ with c colors, the endpoints of this edge cannot have a color in common, so we have: Proposition.7. If a k-fold coloring of a connected graph Γ exists with c colors, then: c = mk if Γ is an m-cliue, c mk if Γ contains an m-cliue. Associated with these multiple colorings is the multiple chromatic number: Definition.8. The k-fold chromatic number χ k (Γ) of a graph Γ for a positive integer k is the minimum number of colors needed for a k-fold coloring of Γ. Such a coloring is called a minimal k-fold coloring of Γ. It is obvious that a -fold coloring of a graph is just a proper vertex coloring. It is easy to see that χ k (Γ) is subadditive: Proposition.9. For each graph Γ and positive integers k, k 2 we have: χ k +k 2 (Γ) χ k (Γ) + χ k2 (Γ). Proof. Let C be a minimal k -fold coloring of Γ and C 2 a minimal k 2 -fold coloring such that C and C 2 have no colors in common. Now each vertex of Γ is colored by exactly k colors of C and exactly k 2 colors of C 2, so we have a (k + k 2 )-fold coloring of Γ and hence an upper bound on χ k +k 2 (Γ). Definition.20 (Fractional chromatic number). The fractional chromatic number χ F (Γ) of a graph Γ is defined as: χ F (Γ) = lim k χ k (Γ) k

14 .2. GRAPHS 3 The existence of this limit is guaranteed by the previous proposition and the following lemma by M. Fekete: Lemma.2 (M. Fekete (923) [20]). If a seuence of real numbers {a n } satisfies the subadditivity condition, then a n lim n n = inf a n n n. For the fractional chromatic number of a graph, this means: χ F (Γ) = inf k χ k (Γ). k We state some properties regarding fractional chromatic numbers: Proposition.22 ([43]). Given a graph Γ we have that: (i) χ F (Γ) is a rational number, (ii) there is a positive integer k such that χ F (Γ) = χ k(γ) k, and (iii) χ F (Γ) Γ α(γ), with euality if Γ is vertex-transitive. Note that the second property means that the fractional chromatic number of a graph Γ is actually the minimum of all χ k(γ) k..2.3 Graph homomorphisms Homomorphisms from a graph Γ into certain other graphs can give information about the parameters of Γ. We will give the definition of a graph homomorphism and show what target graphs give information about the various chromatic numbers. A graph is vertex-transitive if the automorphism group of the graph is transitive on its vertices

15 4 CHAPTER. INTRODUCTION Definition.23 (Graph Homomorphism). Consider the graphs Γ = (V, E) and Γ = (V, E ). A (graph) homomorphism η : Γ Γ is a map from V to V such that adjacent vertices of Γ are mapped to adjacent vertices of Γ. This implies that the fibers 2 over the vertices in Γ are independent sets of Γ. If the map is injective, we call the homomorphism an embedding. A surjective embedding whose inverse is also a homomorphism is called an isomorphism. Two graphs that have an isomorphism between them are called isomorphic. If there is a homomorphism from Γ to Γ we will note this as follows: Γ Γ. If there is an embedding from Γ into Γ the notation becomes: Γ Γ. We will write Γ = Γ if Γ is isomorphic to Γ. Graph homomorphisms can be used to obtain bounds on the various chromatic numbers of a graph. Suppose there is a homomorphism η : Γ Γ, and take a (k-fold) coloring of Γ (for a positive integer k). Now give the vertices of Γ the same color(s) as their images under η. It is clear that this yields a proper (k-fold) coloring of Γ. This shows the following proposition: Proposition.24. If Γ Γ, then χ k (Γ) χ k (Γ ) for all positive integers k. A coloring of a graph Γ with c colors can be seen as a homomorphism Γ K c. Indeed, number the colors m, m 2,..., m c and the vertices of K c v, v 2,..., v c, and define for all vertices x of Γ: η(x) = v i where m i is the color of x. Two adjacent vertices of Γ must have a different color, so they will be mapped to different vertices of K c, and those are adjacent, so η defines a homomorphism. Conversely every homomorphism of a graph into a complete graph can be seen as a coloring of the graph with as many colors as vertices of the complete graph. So we have: Proposition.25. A coloring of a graph Γ with c colors is euivalent to a homomorphism from Γ to the complete graph K c. 2 inverse images

16 .3. FINITE PROJECTIVE SPACES 5 With this in mind we can give an alternative definition of the chromatic number of a graph: Proposition.26. χ(γ) = min{c such that Γ K c }. A natural uestion is now whether we can do the same for a multiple coloring. Take a k-fold coloring of a graph Γ with n colors. Now each vertex is colored by exactly k out of n colors, so consider the graph we will denote K(n, k), whose vertices are all the k-subsets out of the n-set of used colors. We know that two adjacent vertices of Γ cannot share a color, so if we make all pairs of vertices of K(n, k) that are disjoint as subsets adjacent, there is a natural homomorphism from Γ into K(n, k). Again, the converse is also clear, so we have the following euivalence: Proposition.27. A k-fold coloring of a graph Γ with n colors is euivalent to a homomorphism from Γ into the graph K(n, k). As a conseuence: Proposition.28. For each positive integer k we have: χ k (Γ) = min{n such that Γ K(n, k)}. This graph K(n, k) is called the Kneser graph, and we will say more about it in Chapter 3..3 Finite projective spaces.3. Finite fields Definition.29 (Group). A group (G, ) is a set G with a binary operation such that (i) for all a, b G: a b is an element in G, (ii) for all a, b, c G: (a b) c = a (b c),

17 6 CHAPTER. INTRODUCTION (iii) there is an element e G such that a e = e a = a for all a G (this element is called the identity element), and (iv) for each a G, there is an inverse element a such that a a = a a = e. A group G for which a b = b a for all a, b G is called a commutative group. The number of elements of the group is called the order of the group. Definition.30 (Field). A field (F, +, ) is a set F together with two binary operations + and such that (i) (F, +) is a commutative group with identity element 0 (called the additive group), (ii) (F, ) is a commutative group with identity element (called the multiplicative group), and (iii) for all a, b, c F: a (b + c) = a b + a c and (a + b) c = a c + b c where F = F \ {0}. The number of elements of the field is called the order of the field. Remark.3. A set F with binary operations + and that has the same properties as above, except that the group (F, ) is not commutative, is called a division ring. Most of the times we will write just F instead of (F, +, ). Some well-known fields are the field of rationals Q, the field of real numbers R and the field of complex numbers C. Note that they all have infinite order. A field with finite order is called a finite field. The following theorem is a well-known fact: Theorem.32. If F is a finite field with order, then = p h where p is a prime number and h. Moreover this field is uniue (up to isomorphism) with this order. We will denote the finite field of order by GF().

18 .3. FINITE PROJECTIVE SPACES 7 Remark.33. If p is a prime then GF(p) is actually the set of integers modulo p, with the operations defined by performing the operations in Z and taking the result modulo p. This field is also denoted by Z/pZ or F p. Remark.34. The (little) theorem of Wedderburn states that all finite division rings are in fact fields. Definition.35. The characteristic of a field F is the smallest positive integer k, if it exists, such that a + a a = 0 (k repeated terms) for all a F and 0 otherwise. It is denoted by char(f ). It is known that char(q) = char(r) = char(c) = 0 and char(gf()) = p where = p h with p a prime. Definition.36. A subfield (F, +, ) of a field (F, +, ) is a subset F of F that, together with the additive and multiplicative operations of F again forms a field. For example: the field of rationals is a subfield of the field of reals, which is again a subfield of the complex numbers. For the finite fields we have the following: Proposition.37. GF( ) is a subfield of GF( 2 ) if and only if = p h and 2 = p h 2, for some prime p and h h Projective spaces We will introduce projective spaces using their axiomatic description. After that, we will introduce the models of projective spaces that we will work with for the rest of this thesis. First we need to know what a point-line incidence structure is: Definition.38. A point-line incidence structure (sometimes also called a point-line geometry) (P, B, I) consists of two disjoint sets P and B. The elements of P are called points, and P is called the point set. The elements of B are called lines, and B is called the line set. I is a symmetric relation I (P B) (B P), called the incidence relation.

19 8 CHAPTER. INTRODUCTION If (p, L) I for a point p and a line L we can say that p is incident with L (and because I is symmetric, also the other way around). Most of the time we will use the more intuitive expressions the point p lies on the line L and the line L goes through the point p. Similarly, we will say that two lines intersect in a point if they are both incident with that point and that a line connects two points if those points are both incident with the line. With some axioms, we can define a projective space: Definition.39. A projective space is an incidence structure S = (P, B, I) that satisfies the following axioms: (i) Any two distinct points p and are connected by exactly one line (and we can denote this line by p), (ii) for any four distinct points a, b, c and d such that the line ab intersects cd we have that the line ac intersects bd, and (iii) any line is incident with at least three points. Now consider a line L of a projective space S = (P, B, I) and the set P, the set of all points of P incident with L. Define the incidence structure L = (P, {L}, I ), where I is the restriction of I to (P {L}) ({L} P ). It is clear that L is also a projective space, called a projective line. A projective space that is no projective line is called a non-degenerate projective space. A projective plane is a projective space S = (P, B, I) that also satisfies a stronger version of axiom (ii) and a fourth axiom: (i) Any two distinct points p and are connected by exactly one line (and we can denote this line by p), (ii ) any two lines intersect in at least one point, (iii) any line is incident with at least three points, and (iv) there are at least two lines. Note that axiom (ii*) also holds for projective lines, that is why we need the extra axiom (iv). Remark.40. There is an euivalent axiom system for projective planes:

20 .3. FINITE PROJECTIVE SPACES 9 (i) Any two distinct points are connected by exactly one line, (ii ) any two distinct lines intersect in exactly one point, and (iii ) there exist four distinct points such that no three of those points are incident with the same line. Now take a projective space S = (P, B, I). Because two points determine exactly one line, we can identify a line with the set of points incident with the line. Hence we can think of lines as subsets of P. Definition.4. A set A P is called linear if every line meeting A in at least two points is completely contained in A. A linear set containing at least two points must therefore contain at least one line. Denote the set of lines contained in A by B, then it is not hard to see that S(A) := (A, B, I ) is a non-degenerate projective space or a line (here I is the incidence relation I restricted to the set A). The incidence structure S(A) is called a (linear) subspace of S. Like we identified lines with their point sets we can also identify every subspace by its point set and refer to both of them as subspace. It is obvious that the empty set, a singleton of P, a line and the set P itself are examples of subspaces. As every subset of P is clearly contained in at least one subspace, we can define the span of an arbitrary subset of P: Definition.42. The span of a set B P, denoted by B is defined as: B = {C B C, C is a linear set}. It is clear that B is always a subspace. Note that for two sets A, B P we will use A, B as a notation for A B. A set of points A P is called linearly independent if for any subset A A and point p A \ A, we have p A. In other words: A A = A.

21 20 CHAPTER. INTRODUCTION Definition.43. A basis of a projective space S is a linearly independent set of points that spans the entire space. It is not hard to show that every basis has the same number of points. This number is called the rank of S and will be denoted by rk(s). One can define the rank of a subspace in exactly the same way. Take a point p. It is obvious that {p} = {p}, therefore {p} is a basis for the subspace with {p} as point set, hence a point is a rank- subspace. A line is spanned by two distinct points and the set consisting of those two points is linearly independent, so the rank of a line is 2. A projective plane can be shown to have rank 3 and every rank-3 subspace of a projective space can be shown to be a projective plane. The empty set will be considered as a subspace of rank 0. We will define the projective dimension of a projective (sub)space S, to be the rank of S minus one 3. We denote this by dim(s). An m-dimensional subspace (that is, a space of rank m + ) will also be called an m-space. In that way, the empty set is a ( )-space, a point is a 0-space, a line a -space and plane is a 2-space. If the projective space has dimension n, an (n )-space in that space will be called a hyperplane. The following formula is very useful to determine the dimension of the span or the intersection of two subspaces: Theorem.44 (Dimension formula). If U and V are two subspaces of a projective space S, then dim( U, V ) + dim(u V ) = dim(u) + dim(v ). Now we will state two theorems that are important in characterizing projective spaces. Consider a projective space S = (P, B, I) of dimension at least 2. Take any distinct points p, p 2, p 3 and r, r 2, r 3 of S for which the lines p r, p 2 r 2 and p 3 r 3 are concurrent in a point s, and such that no line p i p j or r i r j (for i, j {, 2, 3} and i j) contains s. Define the points 3 The reason for that we want to conserve the intuition that a point has dimension 0, a line dimension, a plane dimension 2, etc.

22 .3. FINITE PROJECTIVE SPACES 2 t ij := p i p j r i r j for i, j =, 2, 3, i < j. Now the projective space S is called Desarguesian if and only if t 2, t 3 and t 23 are collinear for all possible choices of p i and r i. This configuration in a Desarguesian projective space is called a Desargues configuration. See Figure.. Figure.: Desargues configuration This criterion classifies all projective spaces of dimension at least 3: Theorem.45. An n-dimensional projective space with n 3 is Desarguesian. A proof of this theorem is given in [2]. Now take two distinct lines L and M and points l i L and m i M for i =, 2, 3 all different from L M. Define the points t ij := l i m j l j m i for i, j =, 2, 3, i < j. The projective space S is called Pappian if and only if t 2, t 3 and t 23 are collinear for all possible choices of l i and m i. The resulting configuration in a Pappian projective space is called a Pappus configuration. See Figure.2.

23 22 CHAPTER. INTRODUCTION Figure.2: Pappus configuration A connection between Desarguesian and Pappian projective spaces is given in the following theorem which is proved in [26]: Theorem.46. All Pappian projective spaces are also Desarguesian. We will end the axiomatic description here and introduce the model we will work with for the rest of this thesis..3.3 The projective space PG(n, ) Consider the (n + )-dimensional vector space V(n +, K) over the finite field K. More generally one can take a left vector space over a division ring. Define P as the set of -dimensional subspaces of V(n +, K) and B as the set of 2-dimensional subspaces of V(n +, K). Define I to be the symmetrised set theoretic inclusion. Now it is easy to check that (P, B, I) is a projective space of dimension n. This projective space will be denoted

24 .3. FINITE PROJECTIVE SPACES 23 by PG(n, K). It is clear that an (r + )-dimensional subspace of V(n +, K) becomes an r-dimensional subspace of PG(n, ). Thus the rank of a projective subspace of PG(n, K) is eual to the dimension of the corresponding subspace of the underlying vector space, whereas the (geometric) dimension of a projective subspace is one less than the (vectorial) dimension of the corresponding subspace of the underlying vector space. For that reason we will also refer to the (vectorial) dimension of a subspace of the underlying vector space as the rank of that subspace. One can define PG(n, K) in a more practical but euivalent way. Two vectors x, y of V(n +, K) \ {0} are called euivalent if and only x = ky for some k K \ {0}. Now the point set of PG(n, K) is just the set of all euivalence classes under this euivalence relation. The point that is the euivalence class of a vector x will be denoted by P (x), and x is called a coordinate vector of the point P (x). Points P (x ),..., P (x r ) are linearly independent if the corresponding set of vectors is linearly independent in V(n +, K). A subspace of PG(n, K) of dimension r is a set of all points whose corresponding vectors form a (r + )-dimensional subspace of V(n +, ). The reason that PG(n, K) is highlighted here is given by the following theorem: Theorem.47. Let S be a projective space. Then (i) S = PG(n, K) for some division ring K if and only if S is Desarguesian. (ii) S = PG(n, K) for some field K if and only if S is Pappian. From Theorem.45 it now follows that: Theorem.48. If S = (P, B, I) is a projective space of dimension at least 3, then S = PG(n, K) for some division ring K. Note that there exist a lot of projective planes (projective spaces of dimension two) that are not isomorphic to PG(n, K) for some division ring K. If K is finite, it must be a field, according to the (little) Wedderburn theorem, hence K = GF() for some prime power. In this case, the projective space PG(n, K) is also denoted by PG(n, ). But now Theorem.47 states that for finite projective spaces being Pappian is euivalent with them being Desarguesian. From this fact and Theorem.48 one can conclude the following:

25 24 CHAPTER. INTRODUCTION Theorem.49. If S = (P, B, I) is a finite projective space of dimension at least 3, then S = PG(n, ) for some prime power. For the rest of this thesis we will only use finite projective spaces (and hence, over a finite field)..3.4 Some counting in PG(n, ) In this section we will determine some numbers that will play a role in the rest of this thesis. A first useful uestion is determining the number of k-dimensional subspaces in PG(n, ) (where k n). This is the number of rank-(k + ) subspaces of a rank-(n + ) vector space. So let us count the number of rank-k subspaces of V(n, ). We can use the same method that we used in Section. to calculate the number of k-subsets in an n-set. A rank-k subspace is spanned by k linearly independent vectors. For the first vector we have n choices (the all-zero vector cannot be chosen). The second vector cannot lie in the rank -subspace defined by the first, so we have n choices. For the third vector we have n 2. And so on. Finally for the k-th vector we have n k. But the k chosen vectors are not uniue to span this rank-k subspace. So we have to divide by the number of choices of vectors that span the same subspace. Using the same argument this is ( k )( k ) ( k k ). We thus have that the number of rank-k subspaces in a rank-n vector space over GF() is: ( n )( n ) ( n k ) ( k )( k ) ( k k ) = [ n k ] Definition.50 (Gaussian coefficient). The number of rank k-subspaces in a rank n-vector space over GF() is given by the Gaussian coefficient [ ] n k. We can restate this in terms of (projective) dimensions: The number of k-subspaces ] in an n-dimensional projective space over GF() is given by [ n+ k+.

26 .4. FINITE POLAR SPACES 25 Another useful piece of combinatorial information is the number of k- dimensional subspaces containing a fixed l-dimensional subspace in PG(n, ), for l < k < n. This is the same as the number of rank-(k+) subspaces containing a fixed rank-(l + ) subspace in V(n +, ). Now this is the same as the number of rank-(k l) subspaces in the uotient V(n +, )/V(l +, ) which is isomorphic to V(n l, ). So we have that the number of k- dimensional subspaces containing a fixed l-dimensional subspace in PG(n, ) is the same as the number of (k l )-dimensional subspaces in PG(n l, ), which is [ n l k l.4 Finite polar spaces ]. In the previous section we encountered a first kind of incidence structure, namely the projective spaces (and planes). In this section we will define another type: the polar spaces. Definition.5. A (non degenerate) polar space of rank n (n 2) consists of a point set P together with a family of subsets of P, called the singular subspaces, satisfying the following axioms: (i) A singular subspace, together with the singular subspaces it contains is a k-dimensional projective space, for some k n. This dimension is by definition also the dimension of the singular subspace of the polar space. (ii) The intersection of two singular subspaces is again a singular subspace. (iii) Take a singular subspace U of dimension n and a point p P \ U. There is exactly one singular subspace V such that p V and U V has dimension n 2. The singular subspace V contains all the points of U that lie on a common line with p. (iv) There are at least two disjoint singular subspaces of dimension n. Note that an incidence structure that satisfies axioms (i),(ii) and (iii) but not (iv) is called a degenerate polar space. The singular subspaces of (maximal) dimension n are called the generators of the polar space. A finite polar space is a polar space with a finite point set. As with the projective spaces, all polar spaces of rank at least 3 have been classified. In the (thick) finite case all polar spaces of rank at least 3

27 26 CHAPTER. INTRODUCTION are classical polar spaces. Here is a list of all the finite classical polar spaces with their ranks: Q + (2n+, ) non-singular hyperbolic uadric in PG(2n+, ) for some n, giving a hyperbolic polar space of rank n +. Q(2n, ): non-singular parabolic uadric in PG(2n, ) for some n 2, giving a parabolic polar space of rank n. Q (2n +, ): non-singular elliptic uadric in PG(2n +, ) for some n 2, giving a hyperbolic polar space of rank n. W (2n +, ): polar space consisting the points of PG(2n +, ) together with the totally isotropic subspaces of a non-singular symplectic polarity of PG(2n +, ), giving a symplectic polar space of rank n. H(2n, 2 ) : non-singular Hermitian variety in PG(2n, 2 ) for some n 2, giving a Hermitian polar space of rank n. H(2n +, 2 ): non-singular Hermitian variety in PG(2n +, 2 ) for some n, giving a Hermitian polar space of rank n +. The following theorem (see e.g. [28]) gives the size (number of points) of those finite classical spaces: Theorem.52. The numbers of points of the finite classical polar spaces are given by: Q + (2n +, ) = ( n + )( n+ )/( ), Q(2n, ) = ( 2n )/( ), Q (2n +, ) = ( n )( n+ + )/( ), W (2n +, ) = ( 2n+ )/( ), H(2n, 2 ) = ( 2n+ + )( 2n )/( 2 ), H(2n +, 2 ) = ( 2n+2 )( 2n+ + )/( 2 ). In the case of rank 2, there are a lot of non-classical polar spaces known.

28 Chapter 2 Combinatorics and -analogues A natural way of generalizing problems from extremal combinatorics to finite geometry is the -analogue. This is done by changing the definition of the problem as follows: replace all occurrences of the words set of size n with vector space over the field GF() with or rank n. Of course some other words in the definition should also be changed accordingly, for example if the original problem talks about disjoint subsets, the -analogous problem talks about subspaces that intersect trivially. In this chapter we give some examples of such generalizations by taking -analogues of some classical problems in extremal combinatorics. 2. An easy example In Section. we counted the number of k-subsets in an n-set. The -analogue of this problem is counting the number of rank-k subspaces in V(n, ) and that is what we did in Section.3.4. Let us take a closer look at both results. The number of projective points in PG(n, ) is given by [ ] n. If we now recursively define [n + ]! = [ ] n+ [n]! and [0]! = we can write [ n k ] = [n]! [k]![n k]!. 27

29 28 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES Note that this notation looks a lot like the definition of the binomial coefficients. In that light it is not so strange that Gaussian coefficients are also called -binomial coefficients. An even stronger validation of this name is a phenomenon we see with a lot of -analogues. Because we work with GF(), the s used are prime powers. But if we look at the formulas as functions on R in the variable, we can take the limit of these functions for +. For example the limit (by using de l Hôpital s limit rule) of the number of rank -subspaces in a rank-n vector space is: [ n ] lim + n = lim + = n which is of course the number of elements in an n-set. straightforward that [ n ] ( ) n lim =. + k k Using this it is We will see that, in a lot of cases, taking the limit for + of a value in the -analogue of a problem gives us the value from the original problem. Note that this is not always the case; see, for example, Remark 3.9 in the next chapter. 2.2 Sperner s Theorem An important subdomain of extremal combinatorics is extremal set theory. The goal in this domain is to find the maximum size of a family of sets satisfying certain assumptions. In this section, we consider Sperner families Original problem A classical theorem in extremal set theory is Sperner s theorem. If we fix a universe X with size n, the largest uniform family, that is a family in which all sets have the same size, is the family of all k-subsets where k = n 2 (or

30 2.2. SPERNER S THEOREM 29 k = ( n 2 ). This uniform family has size n n 2 ). In 928 Sperner proved ([44]) that this is the best possible upper bound for the size of an antichain in general (for both uniform and non-uniform Sperner families) and that euality occurs exactly in the case just described. There are several proofs known of this upper bound. Here we give one that uses the stronger LYM ineuality. This ineuality is named after D. Lubell [37], K. Yamamoto [48] and L.D. Meshalkin [39]. Here Lubell s Permutation Method [37] is used to prove the ineuality: Lemma 2. (LYM ineuality). If F is a Sperner family of subsets of a set X with X = n, then ( ). n A F A Proof. Consider the subset poset of X: (2 X, ). It is easy to see that the number of maximal chains in this poset is n! and that the number of maximal chains containing a certain subset A is A!(n A )!. Now count the pairs (A, C) where A F and C is a maximal chain that contains A. If we take a subset A in F, we have A!(n A )! maximal chains that contain A. On the other hand, a maximal chain C can contain at most one element of F (because F is an antichain). Thus A!(n A )! n!. A F Dividing by n! gives the reuired result. For other applications of the LYM ineuality, see eg. [33]. Now we can prove Sperner s Theorem: Theorem 2.2. If F is a Sperner family of subsets of an n-set X, then F ( ) n n. 2 Proof. Because ( ) ( ) n n n 2 A for all A F

31 30 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES we have, using the LYM ineuality: A F ( ) ( F ) n n A n analogue Sperner s theorem gives the maximum size of an antichain in the subset poset of a finite set. The natural -analogue uestion therefore is: what is the maximum size of an antichain in the subspace poset of a finite vector space V(n, )? It turns out that the answer is what one would expect, and the proof is a complete -analogue of the set case. We now state and prove the -analogue of the LYM-ineuality. Lemma 2.3. If F is an antichain in the subset poset of V(n, ), then A F [ ] n rk(a). Proof. The number of maximal chains in this poset is [n]! and the number of chains containing a subspace A is [rk(a)]![n rk(a)]!. Counting the pairs (A, C) where A F and C is an antichain containing A yields: [rk(a)]![n rk(a)]! [n]!. A F Dividing both sides of this ineuality by [n]! gives the stated result. If we now use this lemma together with the fact that [ [ ] n k ] n n 2 for all 0 k n we find the -analogue sought for. Theorem 2.4. If F is an antichain in the subset poset of V(n, ), then [ ] n F n. 2 In this case euality holds if and only if F is the antichain consisting of all subspaces of rank n 2 (or n 2 ).

32 2.3. ERDŐS-KO-RADO THEOREM Erdős-Ko-Rado theorem Another classic result in extremal set theory is the Erdős-Ko-Rado Theorem Original problem Theorem 2.5 (Erdős-Ko-Rado Theorem (96) [8]). If F is a k-uniform family with a universe X of size n, where k n 2, and every pair of members of F intersect, then ( ) n F. k A family of mutually intersecting k-subsets of an n-set is called an (n, k)- EKR family. An EKR family that cannot be extended to another EKR family by adding subsets is called a maximal EKR family (or maximal intersecting family). We do not give the original proof here, but we sketch a much more elegant proof by G. Katona in 972 [3], which is inspired by Lubell s Permutation Method. Proof. Consider a labeling of an n-cycle by the elements of X. A path of length k in this labeled cycle corresponds to a k-subset of X. A collection of paths that pairwise overlap has size at most k, and in total there are n paths of length k on this cycle. So at most a fraction k/n of the k-subsets occurring in this cycle as paths mutually intersect. Each subset occurs the same number of times as a path among all possible labelings of an n-cycle by elements of X. So at most a fraction k/n of the ( n k) k-subsets mutually intersect. This bound is tight. We give some examples of families attaining this bound. There is an obvious example of a family of size ( n k ), namely a family that consists of all k-subsets containing a common element of X. Such a family is called a point pencil and the common element is called the center of the point pencil. In Section 2.5 we will show that this is the only type of family attaining this upper bound for n > 2k. Here to overlap means to have at least a vertex in common.

33 32 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES If n = 2k, there are other examples besides point pencils. Each k-subset has exactly one complementary k-subset. An intersecting family can contain at most one k-subset out of each complementary pair. A family consisting of exactly one k-subset out of each complementary pair has size 2( 2k k ) which is the maximum size Analogue In the original Erdős-Ko-Rado theorem, F is a family of k-subsets of an n-set (with 2k n) such that no two members are disjoint. The statement of the theorem can be viewed in two ways:. ( n k ) is an upper bound on the size of F, or 2. F F k k n where F k is the family of all k-subsets of this n-set (this is the statement that was proved by Katona). In a -analogue version of the Erdős-Ko-Rado theorem, F will be a family of rank-k subspaces of V(n, ) with 2k n in which no two members intersect trivially. We will call such a family an [n, k] -EKR family. So now we can expect the following two statements: ( ) F [ n k ], and (2 ) F F k k n where F k is the family of all rank k subspaces of V(n, ). Because [ the ] number of rank-k subspaces containing the same rank- subspace is n k, the ineuality in ( ) is best possible. Furthermore, we have that ] [ n k [ n k ] = k n k n. Therefore ( ) imposes a much stronger bound than (2 ). W.H. Hsieh [29] generalized Katona s permutation method in 973 to prove statement (2 ), and, using a lot of counting arguments, he proved the stronger statement ( ) for n 2k +.

34 2.4. BOLLOBÁS S THEOREM 33 The case n = 2k remained open until 986 when P. Frankl and R.M. Wilson [2] also proved the statement in this case. Examples of families reaching this upper bound are easily found. If n > 2k, a family consisting of all rank-k spaces that contain the same rank- subspace has this size. In the case n = 2k we have the same type of families and also the families that consist of all the rank-k spaces in a given hyperplane (rank-(2k ) subspace). In [2], Frankl and Wilson state they have a proof that those are the only families attaining the upper bound. In a paper, [24], of M.W. Newman and C. Godsil a short proof of this statement is given. Remark 2.6. In [29] and [2], Hsieh, Frankl and Wilson actually prove the following, more general result. Theorem 2.7. If F is a [n, k] -EKR family such that the rank of the intersection of each pair of members is at least t (with n 2k t), then { [n ] [ ] } t 2k t F max,. k t k This result is used to improve some bounds on a similar generalization of the original Erdős-Ko-Rado problem in [8]. 2.4 Bollobás s theorem For all of the problems we encountered in the previous sections, the answer in the original version was the limit case for + of the answer in the vector space version. We already noted that this is not always true. For the following problem we see an even stronger connection between the different versions Set version Helly s Theorem gives a characterization of the dimension of a linear space over R in terms of intersection properties of convex sets:

35 34 CHAPTER 2. COMBINATORICS AND Q-ANALOGUES Theorem 2.8. If C, C 2,..., C m R n are convex sets such that any n + intersect, then all of them intersect. One can wonder if other objects than convex sets obey similar laws. For example, it is easy to see that, if for each set of at most three edges of a graph we have that those edges have nonempty intersection, then all edges of the graph have nonempty intersection. This statement even generalizes over r-uniform set systems (note that a graph is a 2-uniform set system): Proposition 2.9. If each family of at most r + members of an r-uniform set system intersects, then all members intersect. The statement about graphs can also be generalized in a different direction: Theorem 2.0 ([7]). If each family of at most ( ) s+2 2 edges of a graph can be covered by s vertices, then all edges can. This was proved by P. Erdős, H. Hajnal and J. W. Moon in 964. We will not give this proof but prove a more general result later in this section. The complete graph on s + 2 vertices shows that this result is sharp. In 965 B. Bollobás proved ([4]) the following generalization of those problems: Theorem 2.. If each family of at most ( ) r+s s members of an r-uniform set system can be covered by s points, then all members can. We will not prove this theorem, but a slightly more powerful theorem, also by Bollobás in [4]: Theorem 2.2. Let A,..., A m and B,..., B m be subsets of size s of a set of size r such that (i) A i and B i are disjoint for i =,... m, and (ii) A i and B j intersect if i j ( i, j m). Then m ( ) r+s r. There exist a lot of different proofs of this theorem, for example by Jaeger-Payan ([30]) and Katona ([32]). We will give a proof by L. Lovász in [35]. First we need a proposition called the diagonal criterion:

36 2.4. BOLLOBÁS S THEOREM 35 Proposition 2.3. For i =,..., m, let f i : X F, (where X is a set and F a field) be functions and a i X elements such that { 0 if i = j; f i (a j ) = 0 if i j. Then f,... f m are linearly independent in the space F X. Proof. Suppose that m i= λ if i = 0 is a linear relation on the f i s. Substituting a j for some j m in the functions gives λ j f j (a j ) = 0, which implies that λ j = 0. This is true for all j =,..., m, hence the functions are linearly independent. Now we can give Lovász s proof of Theorem 2.2: Proof. Define V = m i= (A i B i ) and associate vectors p(v) = (p 0 (v), p (v),..., p r (v)) R r+ to each v V such that any r + of those vectors are linearly independent (note that there are constructive ways of doing this). Now we associate a polynomial f W (x) in r + variables x = (x 0,..., x r ) to every W V : f W (x) := (p 0 (v)x 0 + p (v)x p r (v)x r ). v W This is a homogeneous polynomial of degree W with: { 0 if x is orthogonal to none of the p(v), v W ; f W (x) = 0 otherwise. Now take a vector a j orthogonal to the subspace spanned by the vectors corresponding to the elements of A j for each j =,..., m. This guarantees that a j is orthogonal to p(v) if and only if v A j. We can see that f Bi (a j ) = 0 if and only if A j and B i intersect, hence if and only if i j. By the diagonal criterion this means that the polynomials f B,..., f Bm are linearly independent in the space of homogeneous polynomials of degree s in r + variables. Since this space has dimension ). ( r+s s ( (r+)+s s ), we have that m