THE MAXIMUM SIZE OF COMBINATORIAL GEOMETRIES EXCLUDING WHEELS AND WHIRLS AS MINORS DISSERTATION. Presented to the Graduate Council of the

Transcription

1 ziq /VQfd THE MAXIMUM SIZE OF COMBINATORIAL GEOMETRIES EXCLUDING WHEELS AND WHIRLS AS MINORS DISSERTATION Presented to the Graduate Council of the University of North Texas in Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY By James V. Hipp, Jr., B.S., M.S, Denton, Texas August, 1989

2 OK Hipp, James, Jr., The Maximum Size of Combinatorial Geometries Excluding Vheels and Whirls as Minors. Doctor of Philosophy (Mathematics), August, 1989, 68 pp., 15 illustrations, bibliography, 16 titles. Ve show that the maximum size of a geometry of rank n excluding the (q + 2) point line, the 3 wheel fg, and the 3 whirl 3F 3 as minors is (n - l)q + 1, and geometries of maximum size are parallel connections of (q + 1) point lines. Ve show that the maximum size of a geometry of rank n excluding the 5 point line, the 4 wheel and the 4 whirl as minors is 6n 5, for n > 3. Examples of geometries having rank n and size 6n 5 include parallel connections of the geometries V^g and PG(2,3). A long line is a line with more than two points. A chain of long lines ^,, ^ has the property that if i j, 1fl = 0 unless i j =1. A ring of long lines I» ^ has the property that i^, 1 2, a chain, and ^ meets only I^ and long lines; W not binary;. The geometry ^ is a binary ring of three is a ring of three 3 point lines that is is a binary rank 4 ring of four 3 point lines; is a rank 4 ring of four 3 point lines that is not binary. The geometry V^g is a rank 4 geometry consisting of two copies of PG(2,3), with three points removed from their line of intersection. The proofs make use of chains and rings of long lines.

3 TABLE OF CONTENTS Page LIST OF ILLUSTRATIONS iv Chapter I. INTRODUCTION 1 Matroids Matroid Minors Uniform, Binary, and Ternary Matroids Direct Sums and Parallel Connections Minor-Closed Classes of Matroids Size Functions and Growth Rates II. THE GROWTH RATE OF THE CLASS OF GEOMETRIES EXCLUDING THE (q+2)-p0int LINE, THE 3 WHEEL, AND THE 3-WHIRL AS MINORS 20 An Upper Bound on the Size Function Extremal Geometries III. THE GROWTH RATE OF THE CLASS OF TERNARY GEOMETRIES EXCLUDING THE 4-WHEEL AND THE 4 WHIRL AS MINORS 30 BIBLIOGRAPHY 67 in

4 LIST OF ILLUSTRATIONS Figure Page 1. PG(2,2) and PG(2,3) 2 2. The k Wheel Graph r 3, r 4, and r v i The Geometry N The Geometry N when k = r(c U 0 = 4 and \l\ = An 8 Point Plane with a Unique 4 Point Line N^, P^, and Vg and Pg A Rank (k+1) Chain of k Long Lines cl(i k _ 1 U = r P = N? P = P An Affine 8 Point Plane 61 IV

5 CHAPTER I INTRODUCTION Matroids Matroid theory is, in part, an extension of affine and projective geometry. Matroids were defined in several equivalent ways by Whitney [13]. Much of our development will follow Crapo and Rota [2]. We begin by defining a matroid using a closure operator, which is used to define closed sets. Geometric lattices of closed sets are next defined. Using geometric lattices, matroid rank is defined. Independent sets in a matroid are defined using rank. Finally, we define dependent sets and circuits in a matroid. Most properties of matroids have meaning on infinite sets, but we will only be concerned with finite matroids. The notation for the set difference of two sets X and Y is X\Y. For a finite set X, denote the number of elements of X by X. For a lattice L, and elements x and y of L, x V y is the join (least upper bound) of x and y, and x A y is the meet (greatest lower bound) of x and y. A matroid is a set M together with a closure operator cl satisfying the following four axioms (White [12, p. 38]):

6 (ell) For every ACM, AC cl(a); (cl2) For every A, B C M, if A C B then cl(a) C cl(b); (cl3) For every A C M, cl(cl(a)) = cl(a); (cl4) For every A C M and a, b G M, if a cl(a U {b})\cl(a) then b cl(a U {a})\cl(a). These four axioms state that a closure operator is increasing, monotone, idempotent, and satisfies an exchange axiom. The collection F of all subsets A of M such that cl(a) = A is called the collection of closed sets, or fiats, of M. If we add the additional requirements (cl5) cl(0) = 0, and (cl6) cl({x}) = {x} for each element x of M, the resulting matroid is called a combinatorial geometry. Important motivating examples for combinatorial geometries are projective geometries of dimension n 1, PG(n l,q), representable over GF(q), where q is a prime power. Figure 1 shows the geometries PG(2,2) and PG(2,3). The flats of these geometries are the empty set, individual points, lines, and the entire geometries. Figure 1. PG(2,2) and PG(2,3)

7 Given a matroid on M, we define the lattice of flats L(M), with elements the closed sets of M, ordered by set containment. For any two flats A and B of M, define A A B to be cl(a n B) = A n B, and A V B to be cl(a U B). A hyperplane H of a matroid on M is a proper closed subset of M, such that for all x e M\H, cl(h U {x}) = M. Define the points (or atoms) of L to be those elements of L which cover cl(0). Let L be a lattice with greatest element 1 and least element 0. L is said to be a point lattice if every nonzero element x in L can be written as the join of points of L. The lattice L is said to be semimodular if, whenever x and y are in L, x and y cover x A y implies x V y covers x and y. If a point lattice L is semimodular and contains no infinite chains, then L is said to be a geometric lattice. It is proved in Crapo and Rota [2, p. 2.11] that the lattice of flats of a combinatorial geometry of finite rank is a geometric lattice, and that each geometric lattice L corresponds to a unique combinatorial geometry, whose points are the points of L, and whose flats are the elements of L. Thus any theorem about geometries of finite rank corresponds to an equivalent theorem for geometric lattices, and conversely. The following proposition gives the necessary property of a geometric lattice to define rank. For proof, see Crapo and Rota [2, p. 2.13].

8 PROPOSITION. If x and y are elements in a semimodular lattice L having no infinite chains, and x < y, then all maximal chains from x to y have the same length. In particular, we define the rank p{y) of y to be the length of a maximal chain from 0 to y. The integer p(y) equals the geometric rank (defined below) of a geometry whose lattice of flats is isomorphic to L. The rank of a point in L is one, the rank of a line is two, and the rank of a plane is three. For the geometry G with lattice of flats L, we define the rank r of each subset X of G by r(x) = In an n dimensional vector space over a field F, a set of vectors v^, Vg,..., v^ is linearly independent if a l v l + a 2 v = 0 implies that each a^ is zero. The rank of a set of vectors V in F n is defined to be the size of a maximal linearly independent subset of V. This notion of rank is used to define a matroid. The rank function r : 2 {0,1,...} is defined by r(a) = p(cl(a)), and satisfies the following three properties: For all X C M, x, y e M, (rl) r(0) = 0; (r2) r(x) < r(x U {x}) < r(x) + 1; (r3) If r(x) = r(x U {x}) = r(x U {y}), then r (X U {x,y}) = r(x).

9 Note that points of a geometry of rank n have rank 1, while hyperplanes have rank n 1. An important consequence of the exchange property (cl4) is that, if M is a matroid and K is a closed subset of M having rank n, then the collection of closed sets in M having rank n + 1 and containing K partition the points of M not in K. Ve will make use of this fact throughout Chapters 2 and 3. A combinatorial geometry on G has the property that the rank of any 2 element subset of G is 2. Thus in a geometry any single element has rank one. The independent sets of a geometry on G are those subsets I of G such that r(i) = 111. An equivalent definition of independent set can be stated in terms of the lattice of flats of G. It can be proved using the proposition above that for each closed set K there is a minimum size subset A such that cl(a) = K. This set A is independent, and A = r(k). The independence axioms for a matroid are: (11) 0 is an independent set; (12) If I is independent, then J is independent for all J C I; (13) If I and J are independent sets with I + 1 = J, then there is an element x of J such that I U {x} is independent- Subsets of M which are not independent are dependent. Define a circuit C of M to be a minimal dependent subset of M.

10 That is, C is dependent, and every proper subset of C is independent. (cl) No circuit is a proper subset of another; (c2) For every two circuits and Cg, if x e flcg, then there is a circuit C (C^ U C2)\{x}. The properties of circuits are satisfied by cycles in a graph, where the edges of a cycle are considered as elements of a circuit, and the edge set of a forest of a graph is regarded as an independent set. In a geometry, three collinear points form a circuit of rank two. Four coplanar points, no three collinear, form a circuit of rank three. A basis of a matroid on M is a maximal independent set of M. All bases of a matroid are the same size. Since the rank of a subset of M is the size of a maximal independent subset, the rank of M is the size of any basis of M. The two defining properties of the collection of bases of a matroid are: (bl) No basis is a proper subset of another basis. (b2) If and B 2 are bases of M and x e then there is a y e 1*2^1 suc h ( ^\{ x }) u {y} is a basis in M. These two axioms are satisfied by bases of a vector space, and by edge sets of spanning forests of graphs.

11 Matroid Minors For a matroid M and T C II, define the submatroid on M\T by cl M^T(X) = [cl(x)] n (M\T) for all X C M\T. It can be checked that cl^rj,satisfies the closure axioms for a matroid. The matroid on M\T is sometimes called the restriction of M to M\T. The set T is the set of elements deletedfrom M. For a geometry G, G\T is called a subgeometry of G, and inherits its geometric structure from that of G, via the closure operator. See Vhite [12, page 131] for further characterizations. A submatroid of a submatroid of M is a submatroid of M. The deletion of the set T from M may be done one element at a time, in any order, always giving the same collection of flats for M\T. For a matroid M and a subset X of M, we define the contraction M/X on the set M\X using the rank function. For each A c M\X, r j[/x(a) = r(a U X) r(x). In terms of closure, contraction of M by X can be defined as cl M^x(A) = [cl(a U X)] \X. The lattice of flats of M/X is lattice-isomorphic to the interval [cl(x),l] in the lattice of flats of M (see Velsh, [11, p. 66]). Contraction of a matroid of rank n by X will produce a matroid of rank n r(x). If x and y are elements of M, then M/{x.y} = (H/{x})/{y} and (M/{x})/{y} = (M/{y})/{x}. Thus contraction of a matroid by a subset X of M can be done one element at a time, in any order.

12 The concept of contraction in a matroid is analogous to that of contraction of an edge e in a graph T. The edges E(T) determine a matroid M(E(r)), whose independent sets are edges of forests in T. Now M(E(r))/{e} = M(E(r/e)), so that contraction in graphs is the same as contraction in matroids (White [12, Ch. 6]). The cycles in T/e are the minimal nonempty edge sets of the form C\{e}, for all cycles C in E(T). This characterization of contraction in terms of cycles in graphs carries over to circuits in a matroid. The circuits of M/X are the minimal sets of the form { C\X : C is a circuit of M }. The contraction of a matroid is a matroid, but the contraction of a geometry by a point is in general not a geometry. Contraction in a geometry is analogous to projection in a projective geometry. For example, contraction of the geometry G = PG(2,3) of Figure 1 by the point x is like the projection of G\{x} from x. There are four lines in G containing x. These lines, being coplanar, represent a 4 point line in the projection. The matroid on G/{x} will have 12 points, have rank 2, and will be partitioned into four sets of three elements, each set having rank 1. One may identify G/{x} with a 4-point line, by deleting eight of the points in G/{x}, leaving a 4-point line {a,b,c,d}. The process of identifying dependent pairs of points and removing any rank-0 elements is called simplification. The

13 simplification of M will be a geometry s(m) whose lattice of flats is the same as that of M. The rank-1 elements of L(M) correspond to the points of s(m). In this work, the contraction of a geometry by a point or line in G will be a geometric contraction. That is, the contraction and ensuing simplification are combined as one operation. When two points x and y, with cl({x}) = cl({y}), are identified, we will customarily delete whichever point is convenient, because the choice of such deletions does not affect the lattice of flats of M. The elements of G/{x} deleted in the process of simplification are said to be destroyed. Ve will write G/x instead of s(g/{x}) for geometric contraction. Throughout this work, The number of points destroyed in the contraction of a geometry G by a point p of G is given by G G/p = 1 + X( / 2), where the sum is taken over the 1 collection of all lines in G containing p. When contracting a geometry by a set X, the contraction can either be done one point at a time, with a simplification after each contraction, or with the simplification taking place after all contractions have been completed. Note that contraction by a rank--0 set is simply the deletion of that set. A matroid on N is said to be a minor of the matroid on M if N C M and the matroid on N can result from a sequence of deletions or contractions of elements of M\N. It can be proved (Velsh [11, p. 65]) that, if x and y are elements of M, then

14 10 (M\{ x })/{y} = (M/{y})\{x}. Thus, if X and Y are disjoint subsets of M, an arbitrary sequence of single element deletions of the elements of X or contractions by the elements of Y can either be represented by (M\X)/Y, or by (M/Y)\X. Uniform Matroids, Binary Matroids, and Ternary Matroids Define the uniform matroid U (m < n) on a set of n m,n v - / elements, by defining its bases to be all subsets of size m. The geometries U2 ^ and U2 g are the 4 point and 5 point lines, respectively. Many, but not all, matroids can be represented as sets of vectors with coordinates from a field F. For example, the Fano matroid F^, which is PG(2,2), can be represented as 7 distinct nonzero vectors in [GF(2)] 3. Matroids that can be represented over GF(2) are called binary matroids. A matroid whose elements may be represented as the edges of a graph (a graphic matroid) can be represented over any field. Matroids which are representable over GF(3) are called ternary matroids. The Fano matroid is not ternary, hence is not graphic. Two matroids of interest to us are the k-wheel and the k-whirl, where k > 2. The k-wheel is the graphic matroid corresponding to the k-wheel graph (See Figure 2). This graph has k edges that are "spokes" and k edges which form a

15 11 "rim", a circuit in Note that has rank k. The k" k complete graph is also the three wheel graph, which is why fg is also denoted as M(K^). The k whirl constructed from 55^ be Jetting the circuits of can be be (1) any circuit of besides the rim circuit; (2) any set of elements formed by adding a spoke of 59^ to the rim circuit of 55^. Figure 2. The k Wheel Graph. The k whirl is not binary, hence is not a graphic matroid. Figure 3 illustrates 5^, 5T^, 35^, and as geometries. The cycles of length 3 in the k-wheel graph evidently correspond to the 3-point lines in The rim cycle in the k-wheel graph corresponds to a circuit of rank k 1 in 5^., while those same points form an independent set in the construction of f 1. Ve will use the convention of only drawing in lines which have more than two points. Since any two points determine a line in a geometry, it is clear what the 2-point lines are in these figures.

16 12 r 3 r- r 4 r Figure 3 Direct Sums and Parallel Connections A proper subset A of a matroid on M is a separator of M if r(a) + r(m\a) = r(m). of M is a separator of M. M is connected if no proper subset Equivalently, M is connected if, for every pair x and y in M, there is a circuit in M containing them both. For further characterizations see Brylawsky [1] and White [12, p. 176]. The importance of wheels and whirls was shown by Tutte [10, p. 1320] (see also Velsh [11, p. 80]). Define a connected matroid on M to be S-connected if the removal of any two elements from M results in a submatroid that is connected. Define an element x of a 3 connected matroid on

17 13 M to be essential if neither M\{x} nor M/x is 3-connected. Tutte proved the following. THEOREM. A 3-connected matroid has every element essential if and only if it is a wheel or a whirl. For disjoint sets N and M and matroids on M and N, define the direct sum M N using the rank function. For all X C M U N, r jj e jj(x) = Tjj(X n M) + r^(x fln). The closed sets of M N are of the form K U L, where K is closed in M and L is closed in N. A circuit of M N is either a circuit of M or a circuit of N. Thus a direct sum is not connected. A basis in M N is the union of a basis in M with a basis in N. See White [12, p. 173] for further characterizations of direct sum. Let M be a matroid which is not connected, with separator A C M. Then the matroid on M is the direct sum of its submatroids on M\A and on A, respectively. Let M and N be two matroids with M fl N = {p}. The parallel connection P(M,N) will have closed sets being those subsets K of M U N such that both K fl M and K n N are closed in M and N, respectively. The main fact about parallel connections relevant to this work relates to contraction. In a matroid on R, if there is a point p of R such that R/p is not connected, then the matroid on R is a,parallel

18 14 connection. That is, there are sets E 1 and E 2 such that U Eg = R., E^ n Eg = {p}, and R/p = [(R\(E 2 \{p}))/p] [(IXCE^p}))/?]. Thus R = P(R\(E 2 \{p}),r\(e 1 \{p})). See White [12, p. 178]. Minor-Closed Classes of Matroids A class of geometries is a collection of geometries closed under isomorphism. A minor closed class of geometries is a collection # of geometries with the property that if G is in tf, then every geometric minor H of G is in The intersection of two minor closed classes of geometries is a minor-closed class. If {G^Gg,...} is a collection of geometries, define the excluded-minor class $a(g 1,G 2,...) of geometries to be the collection of geometries which do not contain any element of p as a minor. All excluded-minor classes are minor-closed. Every minor-closed class can be viewed as an excluded-minor class, simply by excluding the geometries which are not in it. An example of a minor-closed class is %C (q), the class of geometries not containing U 2 q+2 as a minor, where q > 1. Thus U (q) = ^(U 2^+2 ). When q is a prime power, let J?(q) be the class of geometries representable over GF(q). If H is a geometry which is representable over GF(q), then every geometric minor of H is also representable over GF(q). Every geometry H of rank n representable over GF(q) is a subgeometry of PG(n l,q). A geometry representable over

19 15 GF(q) cannot have Ug ^+2 as a minor. Therefore, (q) C U (q). Many characterizations of classes of geometries in terms of excluded minors are known. See White [12, p. 146] for a list. Size Functions and Growth Rates For a class of geometries define the (partial) size function h(n) = max{ G : G #, r(g) = n } if the maximum exists, and is undefined otherwise. Define the growth rate g of by g(n) = h(n) h(n 1) whenever h(n) is defined. There are several results concerning the size function and growth rate of classes of combinatorial geometries. Dirac [3] proved that the number of edges of a graph on n vertices having no topological K^ as a subgraph is at most 2n 3. In Kung [5], growth rate results for various excluded minor classes of binary geometries were obtained. Kung proved that the growth rate of Jf (2) fl &a(m(k g ),F 7 ) is given by g(n) =3, for n > 3, and that the growth rate of Jf (2) fl &(C^Q,M(Kg)) is 4. See [5, p. 839] for a definition of C^Q. Related to the above results are two theorems proved in Kung [6, p. 209]. The graphs Kg and K^ ^ are significant because they are the minimal nonplanar graphs.

20 16 THEOREM. The maximum growth rate of (2) fl &z(m(kg)) is at least 5 and at most 8. THEOREM. The maximum growth rate of =^(2) n &z(m(kg g)) is at least 8 and at most 10. Another line of investigation is to find maximum size estimates for classes of geometries not contained in (2). The main theorem of Chapter 2 is a refinement of a theorem of Kung. Note that is M(K 4 ). THEOREM (Kung [7, p. 233]). When q is odd, the maximum growth rate of 8a(, 3T fl % (q) is at least q and at most q + 1. Vhen q is even the maximum growth of fl It (q) is at least q and at most q + 2. The growth rate of the class &z(s^,5r 3 ) fl U (q) is proved to be precisely q in Chapter 2. Furthermore, the rank n geometries in this class having size equal to the size function h(n) are parallel connections of (q + 1) point lines. In Chapter 3, the maximum size of geometries excluding ^2,5' ^4' anc^ is found. Oxley [8, p. 70] characterized the extremal geometries of Jg (2) fl «a(, and proved that it has size function h(n) given by h(n) equals 3n - 2 if n is odd, and 3n 3 if n is even. Oxley also characterized

21 17 the extremal geometries of =^(3) fl &z( [9, p. 241], and proved that the size function h(n) equals 4n 3 if n is odd, and 4n 4 if n is even. In Chapter 3 of this work, we show that the size function of &a( n U (3) is is given by h(n) = 6n - 5, whenever n is at least 3. This is a partial result to the general problem of estimating the growth rate of H» k ) n v (q).

22 CHAPTER BIBLIOGRAPHY 1. T. H. Brylawski, A Combinatorial Model for Series- Parallel Networks, Trans. Amer. Math. Soc. 154 (1971), H. H. Crapo and G. C. Rota, On the Foundations of Combinatorial Theory: Combinatorial Geometries (Preliminary Ed.), Cambridge, Massachusetts and London, England, The M.I.T. Press, G. A. Dirac, In abstrakten Graphen voustandige 4 Graphen und ihre Unterteilungen, Math. Nachr. 22 (I960), C. Greene, Lectures on Combinatorial Geometries, Notes taken by D. Kennedy from the National Science Seminar on Combinatorial Theory, Bowdoin College, 1971, not published. 5. J. P. S. Kung, Growth Rates and Critical Exponents of Classes of Binary Combinatorial Geometries, Trans. Amer. Math. Soc. (2), 293 (1986), J. P. S. Kung, Excluding the Cycle Geometries of the Kuratowski Graphs from Binary Geometries, Proc. London Math. Soc. (3), 55 (1987), J. P. S. Kung, The Long-Line Graph of a Combinatorial Geometry. I. Excluding M(K^) and the (q+2)-point Line as Minors, Quart. J. Math. Oxford Ser. (2), 39 (1988), J. G. Oxley, The Binary Matroids with no 4 Wheel Minor. Trans. Amer. Math. Soc. (1), 301 (1987), J. G. Oxley, A Characterization of the Ternary Geometries with No M(K^)-Minor, J. Combin. Theory Ser. B, 42 (1987), V. T. Tutte, Connectivity in Matroids, Canad. J. Math. 18 (1966),

23 D. J. A. Welsh, Matroid Theory, London and New York, Academic Press, N. L. Vhite (ed.), Theory of Matroids, Cambridge University Press, Cambridge, H. Whitney, On the Abstract Properties of Linear Independence, Amer. J. Math. 57 (1935), ,

24 CHAPTER II THE GROWTH RATE OF THE CLASS OF GEOMETRIES EXCLUDING THE (q+2)-p0int LINE, THE 3-WHEEL, AND THE 3-VHIRL AS MINORS In this chapter the growth rate of the class of geometries excluding the 3-whirl r 3, 3-wheel 5^, and the (q + 2) point line U2 q + 2 i s found. The main theorem was conjectured by Kung [2, p. 234]. Denote by <U (q) the class of geometries excluding U 2 q+2 as a minor. THEOREM. For q > 2, the maximum growth rate of = U (q) n a( F" 3 ) is q. Moreover, a maximum size rank-n geometry in if has size (n - l)q + 1 and is a parallel connection of n - 1 (q + l)-point lines. A long line is a line that contains more than two points. A triangle of long lines consists of three distinct long lines and such that each pair of these three lines meets (intersects) at a point, and l ± fl l Q f l^ n Zg. It is evident that a triangle of long lines contains a 3 or a W as a subgeometry. 20

25 21 A geometry L is a chain if L is the union of m long lines l ±, l 2,... I m, n <-1 = 1 if i - j =1, and Uj n 1.1 = 0 if i j >2, where 1 < i < m, 1 < j < m. Note that L has rank at most m + 1. Throughout this chapter and the next, we will use the idea of a ring of long lines. If m = 3, a ring is a triangle of long lines. If m > 4, define L to be a ring of long lines if L is the union of m long lines..., ^ (indices modulo m), with the condition that 11, fl /. = 1 if i - j = l or if i - j = m - 1, modulo m, and (. fl ^ =0 otherwise. For a ring, the subscripts {l,2,...,m} are considered to be integers modulo m. Note that L may have rank as high as m. If L is either a chain or a ring of long lines, a point p in L such that p is the intersection of two lines I. and I. is i J called a joint of L. Note that which points of L are joints may depend on how the long lines are labelled in L, because there may be other long lines in L besides the l ±. Define the length of a chain or a ring of m long lines to be m. LEMMA 1. Let m > 4 and let R be a rank m ring of m long lines ip J 2,..., of the l^, 1 < i < m. If iis a long line in R, then I is one PROOF. Suppose that Iis a long line of R, and that I is not one of the Let J = { X[L,x 2,...,x m } be the joints of R, where ^ = l ± fl I m, and = J. n l ± _ ± if 2 < i < m. Let N be

26 22 the points of R which are not joints. Then R is the disjoint union of J and N. The subgeometry J has size m, and is an independent set, since cl(j) = R. Since Ihas at least three points and is a subset of R, /meets at least three of the long lines I L e t L be a minimum length chain in R, such that /fll = 3. That is, L is a chain, and k = 'j ^ 'j+1 U * * U By relabelling the ^ if necessary assume that L = ^ U l 2 U... U / n. Now the rank of L is n + 1, since there n + 1 joints of R in L. But cl(^ U ^ U U = cl(i U l ± U l 2 U... U contradicting the rank of L. 0 LEMMA 2. Let L be a ring of m long lines, with m > 4, of rank m. If y is on ^ and not on l m l or then L/y is a ring of m 1 long lines of rank m 1. PROOF. Write the long lines of L as { i. : 1 < i < m }. Suppose r(l) = m, and let y be a point on which is not on l l or 'm 1* Then the geometry L/y has rank m - 1. Since no other long line in L contains y besides I, only points of I m m are destroyed in constructing L/y. Hence L/y contains at least m 1 long lines. To prove that L/y is a ring, we show that any three noncollinear points a, b, and c in L U L U... U I with i z m 1 7 no more than one of {a,b,c} being on I will have rank 3 in L/y. Denote by r^y the rank function in L/y. From the m,

27 23 definition of contraction we have r L/y({ a > b ' c }) = r ({ a >b»c} U {y}) - r({y}). But for any such a, b, and c, no two of which are on I, m 7 r({a,b,c,y}) = 4. Hence r^y({a,b,c}) = 3 so that a, b, and c are noncollinear in L/y. Therefore L/y is a ring of m 1 long lines, of rank m 1. Now L/y is a ring, since ^ meets in L/y and by Lemma 1, there is no m th long line in L/y. Lemma 2 is related to the fact that if E is a k-wheel (k-whirl), the geometric contraction E/x by a point x of E which is not a joint of R will be a (k 1) wheel ((k - 1) whirl). LEMMA 3. If L is a ring of m long lines of rank m, then L contains a or a as a minor. PROOF. Suppose that for all i < m any rank i ring of i long lines has a or a W as a minor, and suppose L is a ring of m long lines rank m. Since ^ is long, let y e it' y * l n-± n l n> 7 n l ±. Now the geometry L/y has rank m - 1, and consists of long lines L, L, I i ^ in 1, where now l ± n ^ and ^ n ^ are identified as a single point fll n ± in L/y. By Lemma 2 above, L/y is a ring of m a * l n 6 lines of rank m 1, and by induction, H contains ^3 or a as a n

28 24 LEMMA 4. Suppose that 3Z U (q) is a minor-closed class of geometries, n > 3, and for each i, 2 <i<n, He 3 and r (H) = i imply H < (i l)q + 1. If G is a geometry in 3, r(g) = n, and G > (n - l)q + 2, then every point x in G is on at least two long lines in G, and the set of long lines containing x has at least q + 3 points. PROOF. Let x be a point of G. Now G/x is in 3, and by hypothesis G/x < (n - 2)q + 1. Assume G = (n - l)q + 2, since a larger value for G will not decrease the number of long lines through a point. The (n 2)q + 1 lines containing x partition the (n - l)q + 1 points of G\{x}. Let G/x = t< (n - 2)q + 1. Then t points in G\{x} determine the t lines containing x, and there are still (n - l)q t > q points to be distributed among the t lines. These (n - l)q t points cannot be on a single line, because G is in U (q). If the total number of long lines containing x is s, then the number of points on long lines containing x is at least 1 + [(( n ~~ 1)<1 + 1) t] + s, which is greater than or equal to q + 3. D Note that if the geometry G in Lemma 4 satisfies G = (n - l)q + l and every point in G is on at least two long lines, we obtain that each point of G is on long lines containing at least q + 2 points.

29 25 It now follows that for any long line Iin a geometry G satisfying the previous lemma, and a point x on I, the number of points not on Ion long lines meeting I \{x} is at least 2q. This is proved by observing that there must be at least (\l\ l)((q + 2) ( / 1)) points not on Ion long lines which meet I \{x}, which has a minimum value of 2q if \l\ is 3 or q + 1. Also, from the above remarks, it follows that if G has size (n - l)q + 1, r(g) = n, and each point in G is on at least two long lines, this lower estimate becomes 2q - 4 for q > 2. Geometries Having Maximum Size The maximal rank 2 geometry in # = (q) fl F" 3 ) is the (q + 1) point line. Vhen the rank n is greater than 2, there exist geometries of size (n - l)q + 1 in which are parallel connections consisting of n - 1 long lines each of length q + 1. For rank 3, this consists of two (q + 1) point lines which meet. The following proposition shows that these geometries are maximal. PROPOSITION 5. Suppose G is a rank-n geometry in U (q) of size at least (n - l)q + 2, where n > 3, and q > 2. Then G contains a or a as a minor. PROOF. The proof is obvious if n = 2. If n = 3, let a and b be two points of G. Suppose that G does not contain a O

30 26 or a F 3 as a minor. Partition the 2q points of G\{a,b} into those which are on the line cl({a,b}) and those which are not. The former has size at most q 1, and the latter has size at most q. Thus G has at most 2q + 1 points, contradicting the assumption that G has at least 2q + 2 points. Suppose n > 4 and q > 2. Let I^ be a long line in G, with two points x^, x 2 on l^. Now every point in G is on at least two long lines, by Lemma 1, so let ^ be another long line containing x 2. By the remarks preceding this theorem, there are at least 2q points not on 1 2 on long lines which meet ^\{x 2 }. If any such long line Ialso meets l ±, then l ±, ^2> and Iform a triangle of long lines. Otherwise, assume that none of these lines meet l^. Call the collection of long lines which meet ^\{x 2 } K 2 the union of the long lines in Jfy, and T 2 the set l ± U ^ U K 2. Now T 2 > 2q + 5, so that if the rank of T 2 is 3 it follows that G contains a or a T as a subgeometry. If the rank of T 2 is at least 4, then there is a long line meeting / 2 \{x 2 } such that ^ U Z 2 U / 3 has rank 4. Let x^ = fl. Suppose that for i < n the chain L. = L U L U... U Z has been constructed in G, with rank i + 1, where e x 2 = *1 n *2' x 3 = *2 n *3» * > x i = n ^» with K. constructed for each j < i, T. = L U L U... U /. U K.. L z J J 3 Further suppose that for j < i, rank(t.) = j + 1 implies G J o contains a or a W as a minor. There are at least 2q

31 27 points not on I^ which are on long lines meeting Call this collection of long lines and let K i = U{ I: I }. Suppose that a line in JS. meets a line J, 1 < j < i. Let m = max{ j : 1 < j < i and a line in meets I }. J Then a line Imeeting (. and l m, together with, form a ring of i - m + 2 long lines of rank i m + 2. If no long line meeting /^\{x^} meets any of the long lines in L.^, then the subgeometry """i = *1 U ^2 U ' ' * U IIKj forms a rank (i + 1) subgeometry of G. The contraction T^/x^ is a rank i geometry with points Z 2 U J 3 U... U I. U I.. Now U jg U... U is a chain of long lines in T^/x^, the rank of T i /x 1 is i, and by the induction hypothesis G contains a ^ or a W ^ as a minor. Since i cannot exceed n, this proves the proposition. D Ve now prove that any rank n geometry G of size (n - l)q + 1 in ^ is a parallel connection of (q + l)-point lines. For q = 2, see [1]. Assume q > 3. It is obviously true for n = 3 that G is the parallel connection of two (q + 1) point lines. Suppose that for n = 3, 4,..., k that Kg K = (n l)q + 1, and r(k) = n imply K is a parallel connection of n - 1 (q + l)-point lines, and let G be a rank (k + 1) element of with G = kq + 1. If each point of G is on at least two long lines, then from the

32 28 remarks preceding Proposition 5 we have that for any long line Iand a point x on Ithere are at least 2q 4 points not on Iwhich are on long lines meeting I \{x}, and we may use the proof of Proposition 5 to show that G contains a or a T as a minor, a contradiction of the definition of Since each point of G is on at least one long line, there is a point p of G such that p is on only one long line. Call this line I. Now Ihas length q + 1. It follows from the induction hypothesis that there is a second point c on I, such that c is on a long line besides I. Now G/p is a parallel connection of k - 1 (q + l)-point lines by hypothesis. Thus every plane P containing Iin G such that Py > 2 is a parallel connection of Iwith another long line of size q + 1. The geometry consisting of Iand the long lines meeting Iis a geometrically closed subset of G. If it is all of G we are done. Otherwise, consider a plane P containing p such that P meets a long line V which meets I, and P/p = q + 1. If P is not a (q + 1) point line together with p, then I U I' U P will have a ^ or a r 3 as a minor, a contradiction. Similar arguments can be used to show that G is k 1 parallel connections of k long lines, each of size q + 1. This completes the proof of the theorem.

33 CHAPTEft BIBLIOGRAPHY 1. T. H. Brylawski, A Combinatorial Model for Series- Parallel Networks, Trans. Amer. Math. Soc. 154 (1971), J. P. S. Kung, The Long Line Graph of a Combinatorial Geometry. I. Excluding M(K^) and the (q+2)-point Line as Minors, Quart. J. Math. Oxford Ser. (2) 39 (1988),

34 CHAPTER III THE GROWTH RATE OF THE CLASS OF GEOMETRIES EXCLUDING THE 5-POINT LINE, THE 4 VHEEL AND THE 4 WHIRL AS MINORS The main result of this chapter is THEOREM. Let h(n) be the size function of the class # of geometries excluding ^ or as minors. Then h(n) =6n 5, for n > 3. From this theorem it follows that the growth rate of is 6 when n is greater than 3. Examples of rank n geometries whose size is h(n) include the ternary geometries which are parallel connections of PG(2,3) and V lg (see below). Since 2^(3) includes the ternary geometries, these geometries are also extremal rank-n geometries in Jf (3) n T^). The main result is a special case of the general unsolved problem of estimating the maximum growth rate of the class of geometries excluding the (q + 2)-point line, the k-wheel and k whirl as minors. Define V ig to be the rank-4 ternary geometry consisting of two 13-point planes, with three points removed from their 30

35 31 line of intersection. Note that V^g is 3 connected. The ternary geometries without or 2T 4 as a minor, having rank n, whose size is h(n), include PG(2,3), V ig, and geometries which are parallel connections of PG(2,3) or V ig. For rank 4, V^g has size h(4). For rank 5, the parallel connection of two copies of PG(2,3) at a point p has size 25 and does not contain or 5T 4 as a minor. For n > 6, there are rank n parallel connections of copies of PG(2,3) and V 19 which do not have ^ or 5T 4 as a minor, and which have size h(n). Figure 4. V 1Q. It is known that the class ^(3) of ternary geometries has size function (3 n - 1). This is also true of U (3) [1, p. 86]. The extremal geometries of It (3) are PG(n 1,3). Define two points x and y in a plane P to be tied in P if there is a point z in P, not collinear with x and y, such that cl({x,z}) and cl({y,z}) are both long lines. All geometries in this chapter are assumed to be in U (3).

36 32 LEMMA 1. Let P be a rank-3 geometry in U (3), and x and y be points in P. If there are at least five points in P not on the line containing x and y, then x and y are tied in P. PROOF- Ve may as well assume that there are exactly five points in P\cl({x,y}). First, suppose that there is a line Icontaining either x or y, but not both, such that Ihas four points. Assume without loss of generality that I contains x. Since there are at least two points of P not in cl({x,y}) U I, and P/y = 4, y is collinear with one of these points and a point z of I \{x}, and we have the required z. If there is no such 4 point line l f then x and y are each in distinct 3-point lines and l y, respectively, with x not in I and y not in I. If I and I meet, we are y x x y done, otherwise we use that there is a fifth point z which is collinear with y and a point of I. This concludes the A proof. g If P is a plane of size 9 or larger, then for any two points x and y of P, x and y are tied in P, by Lemma 1. LEMMA 2. Let!be a long line in a plane P, such that P\i = 4. Then at most one pair of points x and y in /is not tied in P.

37 33 PROOF. Suppose that Icontains two points x and y that are not tied in P. Since each of x and y is on a long line in P besides I, x is on a line k^ = {x,x^,x 2 } in P, y is on a line k 2 = {y,y 1,y 2 } in p > such that the tw0 lines do not meet. Let z 1 be a third point of I. Now z^ is on a line containing either or x 2, and either y^^ or y 2, since P/z^ < 4. Therefore and x are tied in P, as are z^ and y. If z 2 is a fourth point of I, then z 2 and x are tied in P, and z 2 and y are tied in P. Ve use the fact that x^^ is collinear with each point of & 2 \{y} and either z 1 or z 2 to conclude that z^ and z 2 are tied in P. A rank 4 ring of length 4 contains a rank 4 ring R such that each long line of R has only three points. If the four points of R which are not joints of R are independent, then R is 5T 4. Otherwise R is Thus a rank-^1 ring of length 4 contains either a or a f 4 as a subgeometry. The following lemma generalizes this. LEMMA 3. Let R be a ring of m long lines of rank m with m > 4. Then R contains a rank 4 ring of length 4 as a minor. PROOF. The lemma is obvious for for m = 4. Suppose that the theorem is true for m > 4, and R is a ring of m + 1 long lines, having rank m + 1. Choose an arbitrary point y of R

38 34 which is not a joint. By the definition of a ring, y is only on one long line in R, so R/y is a ring of m long lines, and has rank m, by Lemma 2 of Chapter 2. Let D be a chain of long lines I ^» ^ - 3, r(d) = k + 1, having joints x^ = ^ (2 < i < k)> with x^, points on ^ (respectively) which are not joints of D. Let N be a geometry containing D, such that there are points a, b, c, d in N\D, with the property that { x k+l' a ' b }' {x^^jcjd}, and {x^,b,d} are distinct lines (Figure 5). N Figure 5 LEMMA 4. The geometry N contains a ^ or a as a minor. PROOF. By restricting to a subgeometry of N if necessary, we may assume that N contains no 4 point lines. Ve first prove the lemma for k = 3. Label the points of N as in Figure 6. There are possibly other long lines in N besides those represented in Figure 6, according to the cases which follow. First, if N has rank 5, then we are done by Lemma 1. So suppose that r(n) = 4. The points x^, x^, and

39 35 b determine a plane T. If T contains Xg and yg, then T is an 8 point plane. Now, by Lemma 1, x^ and x^ are tied in T, so that x^, Xg, Xg, and x^ are joints of a rank 4 ring of long lines in N. If T contains x^ and y^, the argument is the same. So assume that T is simply the set {x^,a,b,c,d,x^}. Consider the planes in N containing cl({xg,x^}). There are at most four. Let p! = c1 (( x 2» x 3» x 4})» P 2 = cl ({ x i' x 3' x 4})» Pg = cl({a,xg,x^}), and = cl({c,xg,x^}). From the construction of N and our assumption about T, these four planes are distinct. Therefore, y^ is in one of these. From the construction of D, y^ is not in P^ or Pg. Suppose that y^ is in P^. Then y^ is on a long line in P^, since P^ is a ^ or a N l i yi Figure 6. The geometry N when k = 4 has a ^ or a as a minor. The plane T = cl({x^,x^,b}). There are now four cases to consider. If y^ is collinear with Xg and d, then Xg, x^, b, and d form the joints of a rank 4 ring of four long lines, and we have the lemma. If y^ is collinear with yg and c, then Xg, Xg, y^,

40 36 and y 3 form the four joints of a rank 4 ring of four long lines. If is collinear with y 3 and d, then y 3, x^, b,and d are the joints of a rank 4 ring of four long lines. Finally, if y^ is collinear with x 3 and c, then x^, y^, c, and d are the joints of a rank 4 ring of four long lines. If y^ is in Pg, the proof is similar. This concludes the proof when k = 3. Suppose k > 4, and for j < k, the lemma is true. If the rank of N is k + 2, we are done by Lemma 3, so assume that r(n) = k + 1. Ve divide the argument into three cases. First, suppose that x-^.and y^ are in the plane T determined by a, b, c, and d. Then, in the geometry N/y k, the subgeometry D/y^ is a chain of k 1 long lines of rank k, while P/y^ is a long line containing x^ and x^. Thus we have a ring of k long lines, of rank k, which we know has a f 4 or a F 4 as a minor by Lemma 3. The argument is similar if T contains y^ and Xg. Otherwise, T just consists of x^, a, b, c, d, and In this case, we may contract N by y^. The resulting geometry satisfies the induction hypothesis for k 1. o LEMMA 5. Let C be a chain of k > 5 long lines of rank k + 1, and q a point not in C U cl({xj c _^JX^JX^^}), such that x^ is in cl({xj c _ 1, x k>xk +1,q}). Then C U {q} contains a or a as a minor.

41 37 PROOF. First, if x^, q and a point of either cl({xj c _^) or cl({xj c,xj c+^}) are collinear, we are done by Lemma 3. Otherwise, if x^ either lies in the plane cl({xj c _^jxj^q}) or in cl({xj c,xj c+^,q}), the geometry (C U {q})/q will either contain a ring of k long lines of rank k or a ring of k 1 long lines of rank k 1. If the first two conditions do not hold, then the geometry (C U {q})/q will have x^ in the plane determined by ^, x^, and x^^, but not in the lines *k_l = or = cl ({ x k' x k+l}) * If either of these two lines has four points, then there will be a rank k ring of long lines in (C U {q})/q having x^, x 2,, Xj^, and a point of either or ^ as joints. So assume that these two lines have only three points each. Call the third points of and ^ y^^ an d respectively. If x^ is collinear with Xj^ and a point p of ^ then after contracting (C U {q})/q by p, we obtain a ring of k 2 long lines of rank k 2. Otherwise, x^ is collinear with y-^ ^ and either y^ or x^^, in which case (C U {q})/q contains a ring of k long lines of rank k. If, in Lemma 5, x^ is not in cl({x k _ 1,x^,x^+1,q}), for a given point q not in C, then in (C U {q})/^ the plane cl({xk_^jxj^xj^}) will be the same as in C U {q}. This fact will be useful in the main proof.

42 38 LEMMA 6. Let C be a chain of k long lines of rank k + 1, k > 3, and Q a set of points not in C U c l ^ X j ^ > on long lines containing either x k + i or >v Ifx i or yi is collinear with two points of Q, then C U Q contains a 3^ or a as a minor. PROOF. Without loss of generality we may assume that x^ is collinear with points and qg of Q. Also, we may as well assume that there is no long line in C U Q containing x^ and either or y^, for then we would be done by Lemma 3. Hence, there are points p^ and pg in Q, such that = cl({q^,p^}) and = cl^qgjpg}) each contains either x^^ or y^. Again, without loss of generality assume that contains y^. Ve then have two cases, either that rag contains y^ or In the first case, we are done, by Lemma 4. In the second, first assume that m^ and do not intersect. If cl({q^,q2}) and cl({xj c,xj c+^}) are skew, we are done. Otherwise, we may contract by if k > 4 to obtain an appropriate ring of long lines. If k = 3 and q^, qg, x k, and are coplanar in a plane P, then by Lemma 1 there is a point z in P such that x^, Xg, x,^, and z form the joints of a ring of four long lines of rank four. If ra^ contains y^, m^ contains an d and do intersect, their point of intersection will not be q^ or qg, since by hypothesis there is no long line in C U Q containing either x 1 or y^, and either XJ c+ -J_ or y^. By relabelling if

43 39 necessary, assume that = p 2 - The points x^, q^, q 2, x k+i, p 1? y k, and x k l are coplanar, and form a P 7 (see Figure 9). Thus ^ and x^.are collinear with p^. Thus there is a ring of k long lines of rank k, with joints x^ x 2,..., x^, completing the proof. LEMMA 7. Let C be a chain of k > 4 long lines, of rank k + 1, and Q defined as in Lemma 6, such that either x^ or y^ is collinear with a point of C and a point of Q. Then C U Q contains a or a as a minor. PEOOF. Ve may assume that x^ is collinear with a point z of C and a point in Q, and that there is a second point q 2 in Q such that q 1, q 2, and x k+1 are collinear, by relabelling C if necessary. If we label the long lines of C by I^ = cl({x^,x^+1 }), 1 < i < k, then z is either x^, where 2 < i < k + 1, or z is a point y^ in C which is not a joint, where 1 < i < k. If z is x^ then x^, x^,..., and q^ form the k joints of a ring of k long lines of rank k in C U Q. If z is Xj for i > 3 then x^, x 2,..., x^ are the i joints of a ring M of i long lines of rank i, with the point q^ in cl({xj,xj}) not a joint of M. If z is y^, then similar arguments produce the desired rings.. Finally, if z is x 2 or y^, we contradict the assumption that Q n C is empty. Therefore, by Lemma 3, Lemma 7 is proved.

44 40 LEMMA 8. Let C be a chain of m long lines of rank m + 1 with the usual labelling, and Ia 4 point line meeting C at a point of that is not a joint of C, such that Idoes not depend on cl({x m _ 1 > x m,x m+1 }), but such that I C cl(c)- Then C U 1contains a or a as a minor. PROOF. Suppose that m = 3. Label the points of I by a^, a 2, a^, and a^, so that a^ is x^. If Iand are coplanar, then these two lines meet, and we are done. Otherwise, Iand are skew. Consider the four planes in C U I containing I. Label these planes P ]L = cl(j U {x 3 }), P 2 = cl(z U {y 2 }), P 3 = cl(z U {x 2 }), and P 4 = cl(z U {y^}) (see Figure 7). One of these four planes contains x^. Clearly P^, Pg, and P 4 do not contain x^ since C is of rank 4, and Iis skew with l ±. Therefore, x^ is in the plane P 2, and is collinear with y 2 and a point of I. This point is not a^, because C has rank 4. Since the other three points of I were labelled arbitrarily, assume that x^ is collinear with y 2 and a.^. Therefore the points a^, a^, x^, and y 2 are the four joints of a ring of long lines of rank 4. Suppose that the lemma is true for 3 < j < m, and C and Isatisfy the hypotheses of the lemma, with r(c) = m + 1. If I meets some point of C, then C U I has a ring of four long lines of rank 4 as a minor by Lemma 3. Otherwise, we have that Iis skew with each of the lines 1 2,..., Let P = cl(z m U I). If Xj P, then x^ is collinear with a

45 41 point of Iand x m, and we obtain a ring of i - 1 long lines of rank m 1. If x^ is not in P, but is collinear with a point of C and a point of Ithen C U Icontains a ring of four long lines of rank 4 as a minor, by Lemma 7. If x^ is not in P, and is not collinear with a point of Iand a point of C, then x^ also is not in V = cl(^m_^ U / U /). (This follows from examination of the planes in V containing i> so that x^ would be collinear with a point of Iand a point of C, a contradiction.) If we contract C U Iby x^, the resulting geometry (C U Q/ X i satisfies the induction hypothesis (relabelling the points of C). *1 x. 3 t, y 3 Figure 7. C U Ihas rank 4, and f =4. LEMMA 9. If C is a chain of k long lines having rank k + 1, k is at least 3, C contains two skew 4-point lines, Iis a long line which meets C at a point of i^ which is not a joint of C, Idoes not depend on ^ U fk 1» an<^ ^depends on C, then C U Icontains a or a ST 4 as a minor.

46 42 PROOF. If k = 3, the two skew 4 point lines of C are I^ and Zg. The proof is identical to the case k = 3 in Lemma 8. Suppose that for k = 3, 4,..., m the lemma is true, and r(c) = m + 1. Let and I. be two skew 4 point lines of C, %J i < j 1. If Zmeets any other line in C, we are done by Lemma 3 and Lemma 7. If there is a point p in L = U ^ U... U such that p cl(z U U l^), then by Lemma 5 C U 1 has a or a as a minor. Otherwise, contract C U I by any point p in L\(Z^ U I.) which is not a joint, and use induction. Proof of the Main Theorem For each n > 3 there is a geometry H in = 1C (q) fl 8a( whose size is 6n 5. Thus the size function h(n) satisfies h(n) > 6n 5. For the proof of the main theorem, it will be shown that a geometry G in «(i) whose size is at least 6n 4 will contain a or a as a minor. Assuming that G = 6n 4 is sufficient, since a larger geometry would contain a subgeometry with 6n 4 points. The proof is by induction on the rank n. Cases 1 and 2 below are ad hoc arguments. The proof for n > 6 has two main cases, corresponding to whether or not G contains any 4 point lines.

47 43 Case 1: The Rank 4 Case Suppose that G is in %C (3), r(g) = 4, and G > 20. We will prove that G contains a F 4 or a as a minor. Ve may as well assume that G = 20, since we may achieve this by a restriction. First observe that G/x < 13 for any x in G, since the extremal geometry having rank 3 in It (3) is PG(2,3). It follows that for each point x in G, at least six points (not including x) are destroyed upon contraction by x. Also, each line Iof G is on at most four planes in G. Since at most 12 points are destroyed in a plane when contracting by a line of that plane, each line Iin G is on at least two planes and P 2 such that P^V > 2. Let P be the largest rank 3 subgeometry of G. If IP > 10, then for any point x not in P, there are two long lines containing x which meet P. By Lemma 1, this is sufficient to prove that G contains a or a F 4 as a subgeometry. Suppose that P < 9. Let Ibe a long line in P. By the fact that the planes of G containing Ipartition the points of G not in /, it is clear that P > 8, from the Pigeonhole Principle. Whether P is 8 or 9, and whether l is 3 or 4, there is a second plane P' containing I, such that P'V > 4. By Lemma 2, P U P' contains a ^ or a T 4 ".